What's the point of multiple databases?

I hope I don't regret too much not waiting for morning and a fresh take on
all this.  :-)

So I've got this nice new postgreSQL and it seems to be working nicely.
I've setup a user identity with create database priviliges, and set out to
setup a couple databases for separate projects I want to work on.  That's
what separately named databases are for, right?

Well, maybe it's just a quirk of the utilities (psql and pgaccess), but it
seems that no matter which database I open, I can see and access all the
tables in any database created by this user.  This seems, well, wrong.

This is from the Debian "potato" install, which is based on 6.5.3.  All I
need is a sanity check, please: is this doable, or is there no point in
setting up more than one database per postgres user?

Thanks!