I have set fielddata=true on the type field of the ".kibana" index now, and now I do not get any error when if I query Saved Objects, and there is no Exception in the Elasticsearch logs anymore, so far so good.
Although I can still not use any saved index patterns. Neither through Discover or even see them listed in Saved Objects. I can find the created index pattern object if I query Elasticsearch directly though..
UPDATE: Problem solved!
I did another attempt and instead of setting fielddata=true, set the type of the type field to keyword instead of text, and now it works!
Essentially I had to GET the mapping from Elasticsearch, edit the mapping, stop kibana, delete .kibana, .kibana_1, .kibana_2, then recreate a new Kibana index. After that it works when Kibana is started. Note that this will reset all kibana settings though.