Do I need to mount `elasticsearch-data` volumeClaimTemplates?

According to Volume claim templates | Elastic Cloud on Kubernetes [2.16] | Elastic I have declared the volumeClaimTemplates:

      volumeClaimTemplates:
        - metadata:
            name: elasticsearch-data
          spec:
            accessModes:
              - ReadWriteOnce
            resources:
              requests:
                storage: 10Gi
            storageClassName: local-storage

However, does the manifest need to mount it? If yes, where should be the mount point?

You do not have to mount it explicitly. The ECK operator will add the volumeMount for you.