As stated below

Quote from: https://support.unstoppabledomains.com/support/solutions/articles/48001183211-why-are-some-domains-unavailable-

Minting is the process of pushing your domains from our database onto the blockchain and into your cryptocurrency wallet. Once a domain is minted, the user has full custody over it.

Quote from: jabbathe69 on Today at 11:16:28 AM

What’s stopping someone else from just minting domain names that are purchased by someone else on unstoppabledomains? And what exactly are they doing that prevents someone else from selling NFT domains with their own extensions?

They have their own database a.k.a "intermediary" to detect whether the domain have been purchased or not. So with that, I think that will stop it.

Quote from: jabbathe69 on Today at 11:16:28 AM

And what exactly are they doing that prevents someone else from selling NFT domains with their own extensions?

On the statement above, users can only select the available domain within their database. So, they can't have their own extension.

Quote from: jabbathe69 on Today at 11:16:28 AM

And what prevents me from minting random domain names on Ethereum blockchain? After all unstoppabledomains is not a central authority.

Even if you directly communicate with the smart contract, I believe the contract has its own set of rules to prevent you from randomly minting a domain.