Should the identity of a sender be verified using additional means, or does a MAC suffice?
It is a recurring topic: what exactly is identity and how to prove it ?  But related to your question: a MAC (correctly implemented) proves that the author of the message was in possession of the secret key (like you are, if you can verify it).  Now, depending on the application you have in mind, that's a good enough proxy of identity, or it isn't.

In other words, a MAC takes as assumption that you trust that only the right entities have knowledge of the secret key (you included), and then prove that the message could only have been signed by one of these entities.  The question of whether the right entity has the knowledge of the key is then another issue.