For Azure SQL Database, you can restore any database backup to a new database, and export any existing database to bacpac format on a storage account [0]. The storage account file can be downloaded. You can also do the latter without using through the portal, using sqlpackage.exe on your local machine (same restriction applies: sqlpackage must connect to a live database, so if you want to download/view an Azure backup you must first perform a backup restore).

Not sure how to do the same for Azure Postgresql databases, but looks like standard pg_dump and pg_restore are supported.

[0]: https://learn.microsoft.com/en-us/azure/azure-sql/database/d...