I'm not certain what you are referring to by fingerprints. Bitcoin public keys generally don't use fingerprints like PGP keys do. Fingerprints are used in some descriptors, but that's not really relevant here when generating a multi-sig wallet using seed phrases.

In terms of the derivation path, the answer is maybe. Some people would say yes. I would say that as long as you are using a standard derivation path which will be easy to recover from, then probably not.

For example, I have a multi-sig wallet which was created using Electrum. The derivation paths Electrum uses for its multi-sig wallets is in the open source code and is widely known, so I didn't feel the need to back it up.