Yes, you need to make sure that databases have the same owner, schema changes will travel along the availability group. This gets problematic when there are changes in database owner as you need to reapply those changes too to the secondary nodes.

If it is possible, you could change owner to SA on every database, then make this change every night and when fail-over happens. When you capture the fail-over you can run this to make SA owner for every database:

    EXEC sp_MSforeachdb 'ALTER AUTHORIZATION ON DATABASE::? TO SA';