Getting unlimited amount of signatures - Risk?

Lets suppose we have a communication protocol given where messages $$(1)$$ and $$(2)$$ are send over an unsecure network.

There are some prerequisites like distributed certificates etc., but they should not matter for this question).

Message (1) is send as plaintext from $$A$$ to $$B$$ and does contain the identity of $$A$$. Party $$B$$ responds with a signature over the received identity and some other data (e.g. a public DH key) and sends it over to $$A$$.

A protocol that starts like this design allows an attacker to get a lot of signatures (as much as needed) and the input is even something under his control (the identity of $$A$$).

In my opinion this may open door to a lot of attacks the attacker can use. I don't know any but I can imagine that have been some and this kind of design should always be avoided.

I am open for any kind of comment on this. Especially if there were some relevant attacks in the past.

However, there is one possible avenue for the attacker; if he submits the identity $$Eve$$, and gets the signature for $$(Eve, DHPublicKey)$$, he now has a valid message/signature pair, which would verify just fine. If he can use that signature somehow in an attack, well, that's one thing that the signature method doesn't protect you against. Of course, it's possible that there is no such attack (for example, if the attacker doesn't know the $$DHPrivateKey$$ that corresponds the to $$DHPublicKey$$ in your message), however that is something you need to consider.