Still sounds fishy to me.
Notice when they answer the "are we required to give you the information" they don't actually *answer* their own FAQ question. They just say "well, the FAA is required to obtain this information."
I'd look for something on an official FAA website.
Sounds to me that you aren't required to answer them, but they don't want to do any survey work themselves. I'd also look more into the purposes etc. I didn't see an explanation of a "requirement" to furnish information or even for the FAA to do it. Instead it seemed like the company is trying to entice airports by saying "if you have more based aircraft maybe you'll get more money." Money ends up many times determined by number of operations, not based aircraft. I still think this sounds off and they aren't telling you everything... I wouldn't want the airport giving out my information only know what they've said on that page.