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.

I didn't say it can cache between processes. The problem is not caching between processes, it's that caching itself is not very useful, because the planner creates different plans for different input parameters of the same query in the general case. So you can reliably cache plans only for the same sets of parameters. Or you can cache generic plans, which Postgres already does as well (and sharing that cache won't solve much of the problem too).