There really is such a thing as “senior housing” it is usually housing for people who are over 55. In some cases people who are over 62 years of age. Senior housing can be assisted living but it usually isn’t a nursing home which is a room in a facility with skilled nursing services.
Sometimes I see the term “retirement home”, but not very often I think it is out of style now.
There are also “independent living” facilities for seniors that seem to include assisted living which is housing with services available for a price and apartments or condos in buildings for people over 55 or 62 with opportunities to socialize with other seniors.
I have had some experience working with seniors who have Alzheimer’s. They often can not live on their own. Eventually, they may forget how to use the stove. They may forget how to use the coffee maker and decide that they don’t need to take their prescription drugs and more or they may decide to go for a walk and be unable to find their way back home.
There is assisted living with memory care. It works for people who need a lot of supervision but who do not need skilled nursing care.
Where ever senior lives is senior housing because it is housing a senior. We tend to call old houses historic rather than senior to avoid confusion. There is no rule or law that requires seniors to move. Aging in place is a great option. Not aging is ideal but not an option.