They generally provide assistance for purchasing or improving single family detached housing. There is much less support for other types of housing.
When Americans try to build innovative housing, such as co-housing, they can't find financing for it. This means forms of housing used to provide affordable housing in Europe can only be built in America by groups of wealthy individuals who can afford to self finance. It's also generally harder to build condos and the like because of the lack of financing support generally in the country for anything other than single family detached housing.
Tax breaks intended to make housing more affordable in the big city aren't actually designed to do that. That's not really the outcome you get.
Instead, they are designed in a way that actively encourages people to buy larger homes. This contributes to the size inflation of single family homes and deepens the divide between the have and the have nots. Those who can afford a house at all buy bigger homes, helping to further crowd out those who are poor.
In the 1950s, the average new home was 1200 sqft and housed about 3.5 people. Today, the average new home is around 2500 sqft and houses about 2.5 people, or about one less person.
We are having few children on average, but our houses are larger than ever. We have destroyed about a million SROs and have largely zoned out of existence a variety of smaller home styles currently being called Missing Middle housing.
Some links:
https://en.m.wikipedia.org/wiki/Single_room_occupancy
https://streetlifesolutions.blogspot.com/2018/11/the-clear-c...
https://streetlifesolutions.blogspot.com/2018/12/the-missing...
https://streetlifesolutions.blogspot.com/2018/05/seattle-sta...
https://streetlifesolutions.blogspot.com/2018/05/california-...
https://www.geekwire.com/2018/every-100-families-living-pove...
Street Life Solutions is one of my blogs where I occasionally try to gather data. The 4 posts from it above link out to other resources.