What stops me from owning the same bits of a NFT with a different token?

Nothing. Similarly, nothing stops you from photocopying physical art, or recording streamed music.

The point of NFT is not uniqueness of the item, but uniqueness of the token. Some people see value in having digital proof they bought an item from the artist. Some people see it as rare collectibles.

Regarding the meaning of "the same bits of a NFT with a different token"...

...what I'm understanding is: the NFT is sold on (say) Ethereum. And then later it's relaunched on (say) Cardano.

Did I get that right?

Here's another oddity about NFTs. If you change a single pixel (unnoticeable by the human eye), you get a completely different NFT.

Nothing.