Adjacency works like this:

An improvement gives adjacencies according to the defined types (list at the bottom of the tooltip).

But it receives adjacencies according to its type, which is displayed at the top of its tooltip. Healing pools give adjacency bonusses for approval and influence, but it is only an approval improvement and thus gets no bonusses from a snuggler colony for example.

The Social Matrix for example is a manufacturing and a research improvement and therefore also accepts both adjacencies. However it does not get double adjacency bonusses if an improvement gives both (e.g. antimatter powerplant), it only gets one (the highest one, I guess).

In addition:

the bonus an improvement gets is independent on which kind of adjacency it gets. The bonus it gets only depends on the improvement itself. For example a slave mill (drengin tech) allways gets +5% mnufacturing, +5% research and +5% food, regardless whether it gets it from research or manufacturing adjacencies.

They give bonuses to improvements that have the Influence type, not necessarily to improvements which do anything with influence or which offer the same kinds of bonuses.

Healing Pools have an improvement type of "Approval." Not getting any bonus levels on Healing Pools from being adjacent to things which give levels to Influence improvements is working as intended. Snuggler Colonies have an improvement type of "Trade Resource" and, moreover, don't have any defined bonus for any levels you might possibly get on them, or at least did not last I checked. Not getting any bonus levels on Snuggler Colonies from building things which give bonuses to Influence improvements is likewise working as intended.

Now, if you were to build a Consulate next to a Healing Pool or Snuggler Colony and found that the Consulate did not gain any benefit from being positioned thusly, then there would be a problem, because Consulates are Influence-type improvements and so should receive benefits from anything which gives adjacent Influence improvements bonuses, and both Healing Pools and Snuggler Colonies do so. Healing Pools and Snuggler Colonies not benefiting from being adjacent to one another is not, however, a bug.