I can understand why uncle is reluctant if he's just on vacation from another state. Not going to want to run home to get the paperwork! And it is NOT a federal law.......
As for the "requirement".....our friends stayed at The Palms and never was asked that question (they have 2 dogs). Also, we have 2 cats and have NEVER been asked, anywhere in Oregon, California, Utah, Idaho, OR Arizona for the papers.......and we've been on the road since last fall. If it's something that has come up in the last week, we'd have heard about it in the news.
Call back, and get a different clerk....see if they ask that question again.