The Customer Community Plus license is similar to the legacy Customer Portal: Enterprise Administration license.
So, Community plus license user will be able to update Case
even if its not shared with him (because they can act as delegated administrator).
On other hand, community license user needs the case to be shared with them to be able to update Case
. Check if user has edit access to Case using below query in anonymous window:
System.debug([SELECT RecordId, HasEditAccess
FROM UserRecordAccess
WHERE UserId = '--user_id--'
AND RecordId = '--case_id--']);