Private keys are simply 256bit random numbers. A seed phrase is simply a different way to generate 256bits of randomness in a more human-readable way.
So, to answer your questions
How does the generation of the other 2 Master Keys happen?
You need to securely generate a 256bit random number. This can be done in software. You can also generate them in a HSM. Regardless, for a secure, multi-sig setup, these keys must be generated on different machines, and ideally using different random number generation schemes.
What do these keys share?
For your example, they should share nothing. The user's key and the service/backup keys should be as independent as possible, and hopefully never be active on the same machine.
When new private keys are generated by the user are new keys derived also on the server?
Each master private key must be kept on separate machines, otherwise, you're nullifying most of your multi-signature benefits. Derivations using the private key should only be done when as needed for signing, and the derivations must be independent for each key.
Derivation of the public key can be done on any machine (it's public information). This can help with generating derived multi-sig addresses that can be computed anywhere.