Postgres supports UUIDs natively
Native DB support is irrelevant to me for randomized bits unless it affects storage, sorting, paging, etc. Does it?
You can also write a plpgsql function to generate these ULIDs in the database.
I ran into this last year - storing them as UUIDs works great, unless all of the drivers for the language you're using (Go, in my case) try to cast/validate those bytes as a UUID before you can access them.