No it cannot cache query plans between processes (connections) and the only way it can cache in the same process in the same connection is by the client manually preparing it, this was how the big boys did it 30 years ago, not anymore.

Was common guidance back in the day to use stored procedures for all application access code because they where cached in MSSQL (which PG doesn't even do). Then around 2000 it started caching based on statement text and that became much less important.

You would only used prepared statements if doing a bunch of inserts in a loop or something and it has a very small benefit now days only because its not sending the same text over the network over and over and hashing to lookup plan.

Most systems submit many of the same queries over and over again.

Ad-hoc one off queries usually can accept higher initial up-front compile cost because the main results usually take much longer anyway, vs worrying about an extra 100ms of compile.

Maybe it was too strong to say its not a concern at all, but nothing like PG where every single request needs to replan and potentially jit unless the client manually prepares and keeps the connection open.