Where do we go to find out which NFT is valid according to Beeple? How is this information stored, authenticated, distributed?

A centralized company running a database absolutely can provide digital scarcity. If they could not, then why would there be a real money market around in-game items for many games (which may include some limited run or even one-of-a-kind items).

Sure one can say that NFTs are uniquly identifyable. But that does not mean very much. People might claim it means the owner of the database cannot just make your unique item into something super common by creating many copies.

But they still can create many nearly identical copies differing basically only the NFT token's id, which in effect would act like a serial number.

I don't see this providing any scarcity that a centralized list of "offical" owners does not provide.

The 'NFT company' maintaining the database would still show who 'owned' it long after Beeple is gone. Running such a company could be profitable because you could charge a small fee for the admin: right now it costs around $100 in ethereum fees to buy an NFT, so it could even work out cheaper! As a side benefit, such a company could also host the artwork, ensuring that the inevitable link rot that will otherwise afflict most tokens won't take place. Win win!

How can there be scarcity when anyone can copy the digital files infinitely without permission?

The whole point of artificial scarcity is that something can control and prevent the replication, otherwise it's not scarce.