Mutual Authentication TLS, aka using a client authentication certificate, will only authenticate both parties. The machine initiating the connection ("the client") will receive the server identity certificate, and the machine that had the open port ("the server") will receive the client identity certificate.
Restricting access such that each party allows only the other is an authorization decision, which is something you'd have to do at the application layer (or a library layer that's after TLS but before your program). (Even the notion of "trusted" or "expired" is really an authorization decision, but those are usually default policy that a TLS library will/can apply for you.)
what if a third server (Server C) gets a valid certificate from a Trusted CA and tries to communicate with one of the two Servers A or B
From a TLS perspective, it'll work. From a TLS + default policy (not expired, not revoked, chains to a trusted anchor), it'll work. If you applied custom authorization policy to only allow the A or B identity certificates then you would, of course, need to update that (if appropriate).