No. Not to the other driver's insurance company, not casually, not to be polite, and not because the person who called sounded friendly. Here is why they want it, what they do with it, and what to do instead.
Why they want it on tape
The other side's insurance company records statements for one purpose: to create evidence it can use against your claim. The questions are designed by professionals and asked while you are injured, stressed, and unrepresented. Innocent answers become weapons. Say you are doing okay as a pleasantry, and it appears later as proof you were not hurt.
You are under no legal obligation to give the other driver's insurer a recorded statement. None. The urgency is manufactured.
The traps inside friendly questions
A few classics: questions about speed and distances you cannot actually know, asked so your estimate can be contradicted later. Questions about prior injuries, fishing for something to blame your pain on. Open-ended invitations to narrate, hoping you minimize. And the quiet one at the end, asking whether you are injured, asked before you have even seen a doctor for the full picture.
Adrenaline masks injuries for days. Locking yourself into a same-day description of your condition is exactly what the recording is for.
What about your own insurance company?
Your own policy may require cooperation, which can include a statement. That is different, but it still deserves care, especially in uninsured and underinsured motorist claims where your own insurer's financial interest is opposed to yours. The clean answer: let your lawyer schedule and sit in on anything your own carrier genuinely requires.
What to say instead
Something like: I am not giving a recorded statement. Please direct further communication to my attorney. That is the entire script. You can be polite. You cannot be helpful to them without hurting yourself.
Then call Kyle. Once represented, the calls stop coming to you at all, which is exactly how it should be while you focus on treatment.