Either your password manager will automatically synchronize for you, or you can transfer your passkey to another password manager that will do the synchronization, via the finicky app-to-app transfer system (Credential Exchange Protocol). You can transfer from Apple to Bitwarden and vice versa.
The family sharing question was added later, but the answer is: all of the major password managers have finicky family-sharing features for passwords and passkeys.
For passwords, most people don't bother with formal family-sharing features, and just share passwords via copy and paste. For passkeys, you have to use the family-sharing features, which means you and your family member have to use the same password-manager vendor to share those passkeys.
It’s up to you to decide whether protecting yourself from being tricked into exporting your passkeys is worth sacrificing your ability to read them.