"Fuck you money" is a bad name, then.
But I think it's stuck because, for a lot of people, it is buying a Caribbean island money; it is their chance to say "fuck you" to everyone who still has to work for a living. Their ambition is to display their wealth conspicuously, and otherwise sit on their asses while other people envy them.
I can't imagine someone working because they want to saying "fuck you" to someone who works because they have to, and especially not to the degree that the perpetual vacationer would. In fact, I can imagine the Caribbean island owner being condescending to the willing, but rich, worker because they aren't projecting the image of a rich person. Rich people don't work; but this one does, so he's a misfit.
I think for the crowd that gathers here, "fuck you" money should really be called "next level" money. They likely have that ambition you refer to, where getting rich is just a means to even greater ends. It's hard to quell intrinsic ambition, even if one of the hopes of that ambition has been fulfilled. And it's hard to develop ambition, even when one has the means to become so, but didn't rely on ambition to get there (e.g., lottery winners, etc). This is why most people call it "fuck you" money; it's where they can stop faking ambition and show how little they actually had before getting lucky.