An NFT is a complex abstraction, that can mean many things in many contexts. It's quite powerful as a tool, as a technology. Art is honestly just a fraction of what an NFT can represent. But you won't develop any kind of intuition for that without engaging with it.

As for claiming ownership, yes, if I paid some amount for the ownership of a particular piece of art (commercial rights or otherwise), that is called an exchange or a trade or a transaction. It's quite common in capitalist societies. Not sure why that leaves a bad taste in your mouth. But to each their own.