Many thanks @tsullivan for your reply.
Meanwhile I installed an extra kibana server in 6.8.13 from scratch and I do face the same issue.
About plugin, I noticed contradictory informations in the hereunder logs : "some plugins with status moving to green" and "Discovered 0 plugins"
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:xpack_main@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:graph@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:searchprofiler@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:ml@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:tilemap@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:watcher@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:grokdebugger@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:logstash@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:beats_management@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:index_management@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:index_lifecycle_management@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:rollup@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:remote_clusters@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:cross_cluster_replication@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["info","monitoring-ui","kibana-monitoring"],"pid":29423,"message":"Starting monitoring stats collection"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["license","debug","xpack"],"pid":29423,"message":"Calling [monitoring] Elasticsearch _xpack API. Polling frequency: 30001"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:security@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["status","plugin:maps@6.8.13","info"],"pid":29423,"state":"green","message":"Status changed from yellow to green - Ready","prevState":"yellow","prevMsg":"Waiting for Elasticsearch"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["debug","stats-collection"],"pid":29423,"message":"All collectors are not ready. Waiting for 60000ms longer."}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["debug","monitoring-ui","kibana-monitoring"],"pid":29423,"message":"Skipping bulk uploading because not all collectors are ready"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["debug","monitoring-ui","kibana-monitoring"],"pid":29423,"message":"Resetting lastFetchWithUsage because not all collectors are ready"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["debug","legacy-proxy"],"pid":29423,"message":"\"getConnections\" has been called."}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["license","info","xpack"],"pid":29423,"message":"Imported license information from Elasticsearch for the [monitoring] cluster: mode: basic | status: active"}
{"type":"log","@timestamp":"2021-01-13T09:50:02Z","tags":["reporting","browser-driver","warning"],"pid":29423,"message":"Enabling the Chromium sandbox provides an additional layer of protection."}
{"type":"log","@timestamp":"2021-01-13T09:50:04Z","tags":["debug","config"],"pid":29530,"message":"Marking config path as handled: server"}
{"type":"log","@timestamp":"2021-01-13T09:50:04Z","tags":["debug","http"],"pid":29530,"message":"registering route handler for [/core]"}
{"type":"log","@timestamp":"2021-01-13T09:50:04Z","tags":["debug","root"],"pid":29530,"message":"starting root"}
{"type":"log","@timestamp":"2021-01-13T09:50:04Z","tags":["debug","config"],"pid":29530,"message":"Marking config path as handled: logging"}
{"type":"log","@timestamp":"2021-01-13T09:50:04Z","tags":["debug","server"],"pid":29530,"message":"starting server"}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","http","server"],"pid":29530,"message":"starting http server"}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","http","server"],"pid":29530,"message":"http server running at http://0.0.0.0:5601"}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","plugins-service"],"pid":29530,"message":"Starting plugins service"}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","config"],"pid":29530,"message":"Marking config path as handled: plugins"}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","plugins-discovery"],"pid":29530,"message":"Discovering plugins..."}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","plugins-discovery"],"pid":29530,"message":"Scanning \"/src/plugins\" for plugin sub-directories..."}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","plugins-discovery"],"pid":29530,"message":"Scanning \"/plugins\" for plugin sub-directories..."}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","plugins-discovery"],"pid":29530,"message":"Scanning \"/kibana-extra\" for plugin sub-directories..."}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","plugins-service"],"pid":29530,"message":"Discovered 0 plugins."}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","legacy-service"],"pid":29530,"message":"starting legacy service"}
{"type":"log","@timestamp":"2021-01-13T09:50:05Z","tags":["debug","config"],"pid":29530,"message":"Marking config path as handled: server"}
However I also found this message about the plugin reporting that looks stuck in initializing state
{"type":"log","@timestamp":"2021-01-13T10:15:33Z","tags":["plugins","debug"],"pid":16285,"plugin":{"name":"x-pack","version":"6.8.13","private":true,"author":"Elastic","license":"Elastic-License","kibana":{"build":{"intermediateBuildDirectory":"build/plugin/kibana/x-pack"},"version":"6.8.13"},"resolutions":{"**/@types/node":"10.12.27","@types/react":"16.3.14"},"dependencies":{"@elastic/datemath":"5.0.2","@elastic/eui":"6.10.9","@elastic/node-crypto":"0.1.2","@elastic/node-phantom-simple":"2.2.4","@elastic/numeral":"2.3.2","@kbn/babel-preset":"1.0.0","@kbn/es-query":"1.0.0","@kbn/i18n":"1.0.0","@kbn/interpreter":"1.0.0","@kbn/ui-framework":"1.0.0","@samverschueren/stream-to-observable":"^0.3.0","@scant/router":"^0.1.0","@slack/client":"^4.8.0","@turf/boolean-contains":"6.0.1","@types/json-stable-stringify":"^1.0.32","angular-resource":"1.4.9","angular-sanitize":"1.6.5","angular-ui-ace":"0.2.3","apollo-cache-inmemory":"1.2.7","apollo-client":"^2.3.8","apollo-link":"^1.2.3","apollo-link-http":"^1.5.4","apollo-link-schema":"^1.1.0","apollo-link-state":"^0.4.1","apollo-server-errors":"^2.0.2","apollo-server-hapi":"^1.3.6","axios":"^0.18.1","babel-core":"^6.26.3","babel-polyfill":"6.20.0","babel-preset-es2015":"^6.24.1","babel-register":"^6.26.0","babel-runtime":"^6.26.0","base64-js":"^1.2.1","bluebird":"3.5.3","boom":"^7.2.0","brace":"0.11.1","chroma-js":"^1.3.6","classnames":"2.2.5","concat-stream":"1.6.2","constate":"^0.9.0","content-disposition":"0.5.3","copy-to-clipboard":"^3.0.8","cronstrue":"^1.51.0","d3":"3.5.6","d3-scale":"1.0.6","dataloader":"^1.4.0","dedent":"^0.7.0","dragselect":"1.8.1","elasticsearch":"^15.5.0","extract-zip":"^1.7.0","file-saver":"^1.3.8","font-awesome":"4.4.0","formsy-react":"^1.1.5","get-port":"2.1.0","getos":"^3.1.0","glob":"6.0.4","graphql":"^0.13.2","graphql-fields":"^1.0.2","graphql-tag":"^2.9.2","graphql-tools":"^3.0.2","handlebars":"4.7.6","hapi-auth-cookie":"^9.0.0","history":"4.7.2","history-extra":"^4.0.2","humps":"2.0.1","icalendar":"0.7.1","idx":"^2.5.2","inline-style":"^2.0.0","intl":"^1.2.5","io-ts":"^1.4.2","joi":"^13.5.2","jquery":"^3.5.0","json-stable-stringify":"^1.0.1","jsonwebtoken":"^8.3.0","jstimezonedetect":"1.0.5","lodash":"npm:@elastic/lodash@3.10.1-kibana4","lodash.keyby":"^4.6.0","lodash.lowercase":"^4.3.0","lodash.mean":"^4.1.0","lodash.omitby":"^4.6.0","lodash.orderby":"4.6.0","lodash.pickby":"^4.6.0","lodash.topath":"^4.5.2","lodash.uniqby":"^4.7.0","lz-string":"^1.4.4","mapbox-gl":"0.52.0","markdown-it":"^10.0.0","mime":"^2.2.2","mkdirp":"0.5.1","moment":"^2.20.1","moment-duration-format":"^1.3.0","moment-timezone":"^0.5.14","ngreact":"^0.5.1","node-fetch":"^2.6.1","nodemailer":"^4.6.4","object-path-immutable":"^0.5.3","oppsy":"^2.0.0","papaparse":"^5.2.0","pdfmake":"^0.1.65","pluralize":"3.1.0","pngjs":"3.3.1","polished":"^1.9.2","prop-types":"^15.7.2","puid":"1.0.5","puppeteer-core":"^1.13.0","raw-loader":"0.5.1","react":"^16.6.0","react-apollo":"^2.1.4","react-beautiful-dnd":"^8.0.7","react-clipboard.js":"^1.1.2","react-datetime":"^2.14.0","react-dom":"^16.6.0","react-dropzone":"^4.2.9","react-fast-compare":"^2.0.4","react-portal":"^3.2.0","react-redux":"^5.0.7","react-redux-request":"^1.5.6","react-router-breadcrumbs-hoc":"1.1.2","react-router-dom":"^4.3.1","react-select":"^1.2.1","react-shortcuts":"^2.0.0","react-sticky":"^6.0.3","react-syntax-highlighter":"^5.7.0","react-vis":"^1.8.1","recompose":"^0.26.0","reduce-reducers":"^0.4.3","redux":"4.0.0","redux-actions":"2.2.1","redux-observable":"^1.0.0","redux-thunk":"2.3.0","redux-thunks":"^1.0.0","request":"^2.88.0","reselect":"3.0.1","resize-observer-polyfill":"^1.5.0","rimraf":"^2.6.2","rison-node":"0.3.1","rxjs":"^6.2.1","semver":"5.1.0","squel":"^5.12.2","style-it":"2.1.2","styled-components":"3.3.3","tar-fs":"2.1.0","tinycolor2":"1.3.0","tinymath":"1.1.1","topojson-client":"3.0.0","tslib":"^1.9.3","turf":"3.0.14","typescript-fsa":"^2.5.0","typescript-fsa-reducers":"^0.4.5","ui-select":"0.19.4","unbzip2-stream":"1.0.9","unstated":"^2.1.1","uuid":"3.0.1","venn.js":"0.2.9","xregexp":"3.2.0"},"engines":{"yarn":"^1.21.1"},"build":{"git":{"count":"20611","sha":"2e5162f","date":"Thu, 15 Oct 2020 15:59:15 -0700"},"date":"Fri Oct 16 2020 09:38:03 GMT+0000 (Coordinated Universal Time)"}},"message":"Initializing plugin reporting@6.8.13"}