Can buyers also attempt to monetize their NFT by licensing it out (like an icon or logo)?
No, they cannot. Buying an NFT does not transfer the copyright to the underlying work, and the copyright is what you need to own to license or otherwise restrict the use of the work. In fact, the underlying work is typically hosted at a public URL that anyone, not just the buyer of the NFT, can view. You can even have multiple NFTs created on the same work. In this respect, buying an NFT is similar to buying a signed, numbered limited edition. Other people can still get the regular version of the work at the going rate (which is usually free for online resources), but your version is "special".
As you point out, buying an NFT isn't much like buying a stock. With a stock you are buying a share of a company that will (you hope) make profits. It will then use those profits either to pay out a dividend or to reinvest so as to grow its business and make even more profits in the future. These revenue streams, and the prospects of future revenue, are what people mean when they talk about the "fundamentals" of a security. NFTs have no fundamentals.
Then again, people buy lots of things that have no fundamentals. The traditional art market is a perfect example. So, it's worth asking, how do NFTs compare to investing in traditional art. The main difference is that with traditional art you actually receive the artwork in question. You can hang it on your wall, and if people want to see it, they have to come to your gallery to look at it. (Various kinds of reproductions may exist, like photos or prints, but they're generally of inferior quality to the original. That's why people still go to art museums.) None of that is the case with NFT art sales.
Likewise, with traditional art sales, the way people know you own the work is that they see it hanging on your wall and not somebody else's. With an NFT the only way they know that you own the work is if they go look up who owns it in the registry stored on the blockchain. Right now, basically nobody ever does that. Will they do it in the future? Who can say?
Am I missing something?
No, not really. NFTs are an attempt to replicate for digital art and music the characteristics, especially scarcity, of physical art, and it doesn't really work very well. If your concern is for artists getting paid for their work, platforms like patreon and bandcamp are a better way to do that. If you're looking for a speculative investment, there are lots of things you can speculate on, many of which are more likely to make money for you, and almost all of which have lower transaction costs (selling an NFT typically involves a 10% payment to the originator, plus whatever transaction fees that particular blockchain is imposing). And if you're just looking for a way to show off how much money you have, there are better ways to do that too. At least if you pay to have your name put on a building, people will see it when they walk by.