In HMQV, can one use multiple identities with one public key? The HMQV protocol has 2 "identity" fields $\hat{A}, \hat{B}$ that are used in the dual signature - and it tries to guarantee that any properly-generated session key is only shared by a single $(A, B, \hat{A}, \hat{B})$ 4-tuple.

However, the paper isn't particularly clear whether a party can use multiple, potentially attacker-controlled, identities.
I think you are not familiar with the PKI-based setting. In the PKI-based setting, each user owns a certificate that binds his/her identity with his/her public key. If a user uses multiple identities with one public key, that means he/she must use same public key with a different identity to register many times, at this point he/she will receive different certificates(the number of which is the number of the identities belonging to a user). Well, this is very easy to avoid in the CA, it is only required that repeated registering is not allowed. 

Actually, even if this case happens, then HMQV is still secure only if the secret key corresponding to the same public key isn't compromised.