The short answer is no, don’t give a statement to the other driver’s insurance company or even to your own insurance company.
You may want to believe that the insurance company will be reasonable and pay what it should, but that is simply not the case. In asking you to provide a recorded statement, the insurance companies are hoping you will say something that will give them an excuse not to pay or to pay less than you deserve. They may ask you to explain the accident multiple times and then put all of your answers side by side, under a microscope. If there are any minor differences, they will accuse you of changing your story.