The answer is almost always no. But, why you shouldn’t is more complex: At a certain point the other insurance company is going to be entitled to some portion of your medical records in order to approve your claim. That’s normal and a part of what your lawyer is going to help you with.
I have seen releases that go back to the patient’s or injured person’s date of birth when they were a tiny infant, and demanded all the records from that point forward. Nobody has to give up that much privacy in order to make an auto case, and you still have the right to some privacy.
I don’t recommend giving them your medical records unless your lawyer has figured out exactly the right amount of the records are and answers accordingly.
Talk to an experienced car accident lawyer before you comply with that request.