I did this once, but it was close to 15 years ago; we paid for a rental for my husband's mother and brother to use for our out-of-town wedding. We had to show up together to pick up the rentals, and the driver of the second car had to be vetted in all respects, driver's license, minimum age, etc.
I know that blind travelers often pay for rentals and have someone else drive, so it is possible for the person paying to not be the driver, but that's usually a single-vehicle situation.
I think that you *can* rent and pay for two vehicles in your name, but I'm pretty positive that you will have to go to the counter in such a situation; there is bound to be extra paperwork, and they do have to have identifying information on the driver of the second car. Call National; they will be the ones who can tell you if it is possible (and it wouldn't hurt to get them to send you an email confirming the specifics if it is; the folks at the desk might not be too familiar with the procedure to be followed.)