ECB is perfectly secure when you use it on a single block.
And you have no problem with the same data being encrypted being identifyable (no salt). i see now why it might be useful in this case, though I still don’t like the idea, it feels bad somehow. (Yeah I get that you save storage for some computation)
Encrypting an internal id with ECB into an external id continues to allow the comparison for equality of 2 ids, to determine whether they are the same or not, but except for this it removes all the information contained in the structure of an UUID.
You would still use a secret key, so it's impossible for the end user to decrypt it.
You are encrypting a single block of unique information. No other encryption mode gives you any advantages whatsoever.
Because the internal ID exposes timing/sequence information, as per jonhohle's comment.