@Assaf L Thanks for posting your question in the Microsoft Q&A forum.
That's correct. Unfortunately, there is no option to perform managed backups of partial data from a Postgres flexible server. according to the documentation, Azure Database for PostgreSQL - Flexible Server automatically creates server backups and stores them in zone-redundant storage or locally redundant storage. These backups include the entire server, encompassing all databases created, and have a default retention period of 7 days.
However, you can use the pg_dump tool to manually take a backup of a specific database as mentioned in the FAQ.
Regrads
Geetha