Easy way to find out what tag groups a tag is belonging?

A query that lists tags and has a column of all their associated tag groups?

That should be possible with something like this:

SELECT 
    t.name AS "Tag",
    STRING_AGG(tg.name, ', ' ORDER BY tg.name) AS "Tag Groups"
FROM tags t
  JOIN tag_group_memberships tgm ON tgm.tag_id = t.id
  JOIN tag_groups tg ON tg.id = tgm.tag_group_id
GROUP BY t.name