Maybe it's similar to adding a new chat participant, except your client says "btw this is also me." I used that way when designing a toy e2ee app a while back. Or maybe there is a central repo of each user's per-device pub keys, and your client signs its updates to it using the original pub key so it's tamper-evident.

Maybe there's some key derivation mechanism so the new pubkey is self-evidently owned by the first one, never heard of one though.