It looks like Ubuntu 24.04 no longer includes "legacy" time zones by default. However, I was able to add them back by installing the tzdata-legacy package with
sudo apt install tzdata-legacy
After installing, Postgres correctly recognized things like "US/Pacific" and "Asia/Saigon" again.
It looks like older versions of the tzdata package included a link from Asia/Saigon to Asia/Ho_Chi_Minh. The link is present in 2024a-0ubuntu0.22.04.1 (Jammy) but has been removed in 2024a-3ubuntu1.1 (Noble).