What is the "login with NFT" option?

Wouldn’t it still point to the owner address? Can you please explain how this is a different identity

Login with NFT will provide different identity than owner address. This identity will have it’s own email address and (probably) dedicated mailbox.
After transferring NFT, the new owner will be able to use it to login to various services and will effectively take over this identity.

I don’t have the details nailed down yet so this is only a initial thought.

This here would be great. This could be used similar to NFT platforms such as Rarible and Cargo that provide locked content only the NFT owner can unlock/download. Also Digitible uses an NFT to allow access to enter their building in Decentraland. Only users who have the token in their wallet can enter the building.

What is needed to implement this in such a way in which users can have access to a specific page or stored files only if they have a certain NFT in their wallet?

You would need to use some software that supports OpenID Connect (e.g. OwnCloud https://doc.owncloud.com/server/admin_manual/configuration/user/oidc/) and configure cryptoauth with token restrictions, this unfortunately has to be configured manually.

You can also do the same with discourse, for example there is https://flynnjamm.cryptoverse.cc/ which only allows owners of JAMM token (it’s ERC20). The mechanics would look similar for NFTs but you could configure additional permissions for given NFT token (depending on integration code)