Ah my team mate has spotted the error for me.
And it paid off, that he has no experience with PostgreSQL yet :-)
Because the admin user for Azure Database for PostgreSQL flexible server is
not "postgres" but "postgresql"
Now the connection from a container in AKS works for me:
$ psql "postgresql://postgresql:password12345!!@
172.21.32.4:5432/postgres?sslmode=require"
psql (17.4, server 16.6)
SSL connection (protocol: TLSv1.3, cipher: TLS_AES_256_GCM_SHA384,
compression: off, ALPN: none)
Type "help" for help.
postgres=>
I must find now how to add a non-super user to it next.
Best regards
Alex