Regardless of access modifier, non-static attributes in an Apex class are specific to a given instance of that class.
Now some things to understand about Batchables:
Calling Database.executeBatch simply serializes your Batchable implementation instance into the database.
At some later point the Batchable is deserialized from the database (in a new session**)...
We experienced the penalty delays for chained Queueables in the past (although documentation is rubbish with regards to this). We have 8-10 chained Queueables at times and the AsyncApexJob entries would show a ~60s delay between Completion and Creation but we have not seen these delays for a while now (I would say at least since 1 year ago).