Does Nostr have any mechanisms to ensure anonymity, or can the relay log everything?

> Does Nostr have any mechanisms to ensure anonymity

Your identity on the network is a public/private key pair. You don't have to associate your real name with your posts, if you don't want to. And you can have multiple key pair identities if you want.

Posts are plain text by default, but encrypted private messages can also be exchanged.

You can choose what relays to post to, and obfuscate your ip address with a vpn or tor if you desire.

> can the relay log everything

Yes, subject to the above mitigations.