mortgages in the US are 30 year loans so it stands to reason that most homeowners are still paying their mortgage. The thing is that home values "typically" go up so you can always sell for more than you bought so, yeah, you're in debt but it's secured by an appreciating asset. Further, assuming a couple of other things, inflation makes your mortgage payment effectively less and less as time goes on meanwhile your home is worth more and more.
> but surely this number includes people with mortgages that haven't yet been fully paid off?
It definitely does, but most people don't lose their homes. Even in 2008.
Actually the swing from the absolute high of 2005 to the low of 2016 is only ~6% of the population owning their homes (never goes below 60%).
Longer term graph: https://fred.stlouisfed.org/series/RHORUSQ156N
Now, might some people have to downsize? Absolutely. Some people are absolutely living beyond their means. That being said, mortgages are pretty safe. If you get one, there's a real good chance you will end up owning your home at some point. The problem with mortgages in 2007/2008 were that people were taking higher risks and trying to chop up the numbers in a semi-fraudulent way (and in several cases, blatantly fraudulent).
If this is the case, then to me this data simply illustrates that up to 2/3 of American households were able to obtain bank loans. Perhaps I'm being too cynical, but recent history has shown us that being able to obtain credit doesn't necessarily mean you can afford the underlying thing.