Two-way secret sharing for authentication?
This scheme is no different from standard password-based authentiation / standard cookie-based authentication:

You have a token (call it "password", "cookie" or "share") and an ID (call it "username"), you send it to the server, the server does some processing using additional values only it knows and gives you a yes / no answer for whether or not you are authenticated.


For classic passwords, this would be password-hashing with a DB-stored salt and comparison to the stored salt.
For cookies, this would be either a static comparison or an integrity check on the cookie server-side with a server-side static key.
For the given scheme, this is the secret-recovery and verifying whether the recovered content matches a specific format.



  Is this authentication scheme secure?


It is as secure as the standard approach of sending a username and a password. So it usually works, especially if your channel is authenticated, eg using server-authenticated TLS, anyways.


  If an adversary were to intercept a secret share for a user, would
  anything other than the user be compromised?


Well, if the server-side shares are user-dependent and the shares are unrelated / independent, then obviously nothing can happen to the others if one is compromised. The exception being of course if the user in question can log-in as an admin or something and just read-out or modify the shares or the received shares.