Postgres has session-bound resources, and in most cases no way to disable those from being deleted when exiting a session. For example in postgres you can't persist a prepared statement, but you can of course persist data within a table. Any function running will be killed when you exit (or at least not complete since the transaction is cancelled).

IMO when a user has logged out and has not had the permissions/foresight to setup a task in the system to run without a session it should be killed.

I get that this has not been the default behavior in linux/UNIX, but to me it seems like the sensible one.

And that's before we ever argue about the possibility to turn it off.