I was just looking into this. Was trying to get a sense of the number of total US housing units compared to the number of households. And the number of extra units seems to have been fairly constant over the last 20 years (about 9% more housing units than number of households).
Yes, today there are roughly 1.1 housing units per household, which is exactly in line and consistent with history. This is also trending up, as population growth slows and rate of building increases
Shortage narrative is propagated by people who don't look at the data
[1] Total US Households: https://fred.stlouisfed.org/series/TTLHH [2] Total US Housing Units: https://fred.stlouisfed.org/series/ETOTALUSQ176N#0