I don't get it. Is this exclusive to Powerplay? Does this mean that if my faction is a democracy, the galmap will show the system as a democracy even if the controlling faction is an anarchy?
Yes, if most of the anarchists assets in foreign systems are subjected by you, the democracy.
For example:
Khemaraui ruled in its home system and the two colony systems nearby. Otohime subdued them. Now 3 of their main assets are below Otohime in Khemaraui, but they still own 2 in the others. Still those two were declared communists as well.
At that point we thought it might only be related to "colonies" if all your main systems are beaten.
Then came ToC:
After they lost the rule in HIP 4273 to the Communists, HIP 1672 and Synuefe GU-D D13-78 both turned communist. If it had been part of the "homeworld control" it would be patronage as Vesuvit was reconquered by the Patronage. But instead it turned communist. Yet two systems Toc is in are "ruled" by communists where they have assets (4273 and Kureserians). Other systems they are in also have partially communist governments (6369). So it seems their "government trigger" flipped to communism as most of their overlords are communist.
I don't quite understand the mechanic either, but it clearly resembles medieval feudalism. If you own assets in other systems of government without ruling them, you get "infected".