How to get actually allocated tablespace sizes?

A Postgres tablespace is cluster-wide.

Therefore, how do I calculate how much space is allocated by each database in Postgres tablespaces?

To get the disk space occupied by a database (nicely formatted):

SELECT pg_size_pretty(pg_database_size('mydb'));

Disk space occupied by a tablespace:

SELECT pg_size_pretty(pg_tablespace_size('mytblspace'));

Find out more about database object size functions in the manual:

StackExchange Database Administrators Q#14623, answer score: 20