At the end of the day "ownership" mostly has any meaning within the legal system. NFTs seem to create something akin to a shadow copyright but which makes no sense. Oh well, maybe I am just a simpleton.
No, no, no. It's worse than you think. An NFT is more like a plastic cup with Pikachu printed on it. You don't own the rights to Pikachu. You just bought a cup with his face on it. With NFTs, there is no contract to own the actual "thing". Just simply the piece of "merch" (the NFT) with the likeness associated with it. Except, a plastic cup still has value to drink from or hold your toothbrushes.
E.g. on some art NFT platforms, a legal license is attached that says that if you hold the NFT for an artwork, you get permission to display that just as you would have permission to display lets say a piece of software art you bought from them. The NFT just is a representation of who holds that license right now, instead of a software license represented by a piece of paper being sold, or an entry in a license database somewhere.
Digital artists have struggled with how to sell a digital thing that can be copied losslessly for a while. Expensive digital art is often sold as some physical thing, e.g. a print, an installation with some screens showing the artwork, or a movie on DVDs. The limited number of the physical things provides a scarcity that apparently has value to a lot of people, leading to higher sales prices, but is also somewhat frustrating if you want the digital thing to be the art. In contrast, when they try to sell just files, that's basically not happening. People will spend hundreds on a print, but not $5 on the digital file the print was created from. And for some, it now turns out that NFTs also provide that scarcity, in the purely digital domain. Sure, everyone can go and download the file, but if you bought the NFT you have a record that proves that you actually bought the thing from the artist. That appears to have value to some people, and they pay the same kind of money as they would for a print etc.
If you don't have an explicit legal thing, it comes down to social convention that this (virtual) thing you are holding has some value to some people. Some collectibles are somewhat of an analogue: Stamps. signed photographs. pokemon cards. Skins in Counter-Strike. All those have no obvious reason why they are worth a lot a money (or not), except that enough people want them and are willing to spend to get them. If a piece of paper with some dudes signature somehow has value to some people, why can't a bag of bits with a digital signature have value to others?
And of course there's lots of hype and too much money floating around that makes all this more extreme, leads to people throwing money at it in the hope to make a profit, and makes the amounts paid for the headline-making things just seem entirely stupid. But that's roughly the forces at work, if we agree to the specific value judgements or not (very similar to collectables markets again, those seem equally weird to me in the extremes).