It's more like co-owning. Both the owner and the HOA own the property. But at least with my HOA the agreement is they handle everything outside my walls, I handle everything inside.
HOA is just a name, they take many many forms. My grandparents neighborhood for example has an HOA that collects dues and the dues are used to pay to have the roads repaved or fixed when damaged, and that's it. No upkeep rules or anything like that, and there's only a handful of houses in their small neighborhood that look bad from the outside, but that's those people's business
Depends who owns it / who payed to have it constructed. If the tenets own it and payed to have it constructed, or payed someone who payed someone... Then it's a condo or a co-op depending on whether you own a unit in the building or own x% of the building which entitles you to a unit.
If another organization, almost always some form of government, payed to have it constructed and owns it then it is public housing.
If it's anything like cities skylines original, houses just pop up according to demand with seemingly no construction cost calculated, probably because it would add a ton more complexity with mortgages and speculative markets etc. for little gain to players who mostly just want to play with trains and metros.