Earlier quoted context omitted.
Given that a UUID identifier fits in a single cipher block, and the whole point is that these are unique by construction (no IV needed so long as that holds true), it seems like a single round of ECB-mode AES-128 would enable quickly converting between internal/external identifiers. 128 bits -> 128 bits
That's an interesting idea, how would you deal with the bits in the UUID that are used for the version? Setting them to random bits may cause issues for clients that try to use the identifier in their own database or application, as mentioned in the article.
Alternatively, just say it's a random string ID and not an UUID.