Disclaimer: Out over my skis again.
From what you say here, and over on SO, it sounds like you've got two problems:
* Matching on huge numbers of records because of common tags.
* A dynamic collection of tags as they're customer driven/configured.
An "ideal" solution might look like a bit-index for each tag+tuple, but Postgres does not have such a structure. The closest I've seen are Bloom filter based indexes. That's likely not going to work here as you don't know the collection of tags at any one time. If, however, you create your own frequency count estimates for tags, you may well find that there are a small number of common tags, and a large number of rare tags. That would be good to find out. If you do have some super common (non selective) tags, then perhaps a Bloom index based on that collection could be effective. Or expression indexes on the very common tags. In your SaaS setup, you might need counts/indexes tied to some kind of customer/tenancy distinction ID, understood. But, for simplicity, I'm just saying a single set of frequency counts, etc.
Here's a recent article on Bloom filter based indexes in Postgres that looks decent: