There'll often be a broad restriction of adornment of any kind outside of a strict list, and/or at the discretion of the HOA/property manager. Many don't allow window AC units. There's a general air of paranoia about anything that could potentially bring down perceived property values, or that might otherwise project a sense that the neighborhood is anything other than a Flanderization of affluence. (There's also a element of social control.) Think historical preservation codes, but for a pile of sticks built in the 80s or 90s.

Ah, I see.

For the whole "land of the free" and "free market" thing, the US seems very not that?

Well, the US is a strong federal state, so it depends on the level of government. At the national level, the US government is relatively hands-off compared to other places. At the local level, it depends on your local politics. In urban areas, you might have an HOA telling you whether or not you can have an AC unit. In rural areas there's almost certainly no HOA, and potentially not even a local municipal government at all, and could quite often be legal to put up a gun range in your back yard.

Unpopular opinion: Things here get twisted by our sordid history with race/class. We actually do value our civil liberties and economic freedom, as a general rule... but that can and does get short-circuited by attitudes and assumptions that were steeped during segregation and industrialization (and the associated widening of economic inequality).

Our government had the bright idea to bake those issues - particularly the strict rich/poor, white/black, good/bad dichotomy - into our housing policy, so now, any divergence from the local (affluent) norm isn't just a funny quirk; it conjures up anxieties associated with the Civil War, white flight, immigrant ghettos, eminent domain, urban decay, Superfund sites, etc.

People here are desperate not to be on the wrong side of the tracks, as it were, and so they'll submit themselves to no small amount of what looks like insanity to the rest of the world, in order to not live somewhere thought of as "sketch". Not entirely irrational, mind you, since these kinds of perceptions are often what determines whether or not a neighborhood receiving amenities like "parks" and "school funding" and "a place to buy food."

Circling back: window AC and PV signal to some people that folks in the neighborhood are too poor to afford central AC or roof panels (or to not "need" solar, budget-wise). These people (and the people who want to sell their homes to the first group) will fight you to prevent that perception from taking root. It wouldn't be as much of a problem if so much wealth wasn't tied up in real estate (the buildings, not just the land), but that's where we are.