People have to live somewhere. If you're renting and this insane policy gets enacted, you now need to buy something or you'll be out on the street. So supply increases, but demand increases by the same amount.
if everyone was limited to 1-2 units nationally (let’s say 1 home/unit in high demand cities), then there would absolutely be more supply than demand.
A 400 unit apartment building where the landlord legally must sell 399 units (or all 400 assuming they have other properties), repeated en masse throughout the city, would truly see there being more units on the market than buyers, because a) investors and high net-worth individuals would not be able to buy, as they would be having to sell off to get under the legal cap b) the only available buyers would thus be those who are under the limit of the housing cap, i.e. renters, which are mostly those who could not afford a home at the previous prices in the area
Considering the legal onus would be on the previous owners to sell per this insane law, sellers would rather make $100k>$10k>$1k than $0 and so there would be a rush to sell, exherting immediate downward prices on homes.
In San Francisco, which is a city where >66% rent where they live (and thus <33% of the city owns the whole housing stock), you would see a giant flood of homes hit the market that the existing landed gentry would be unable to snap up, equalizing around a 100% live-own rate (because, again, once you buy one unit you can’t buy another, no matter how cheap it is)