If you connect but don't auth, the relay doesn't get an association between your npub and your ipaddress.  They just know that some IP address connected to them, but not who it is.  Of course, once you start interacting with the relay they learn all of that too, and maybe the relay can try to figure out who you are by which posts you are interested in... and if you post they might presume you are that event's author.  But you might be transmitting somebody else's message too, so it is not proof.  So it gets all very complicated as to how much a relay can learn.

But if you AUTH it is not complicated at all, it is pretty much just giving the relay a provable association.

I've always maintained that privacy is an illusion unless you use tor, and trying to hide IP addresses is just more pretend privacy that enhances people's false sense of privacy.