That's 65% of homes are occupied by the owners, not 65% of the population owns their home.
Those are wildly different things and people keep using the former in a likely attempt to portray the latter as much higher than it really is.
In fact, that second statistic is impossible to find as every article or study from a mainstream or even semi mainstream source uses only the misleading one.
Tl; dr:
"How many homes are occupied by their owners" ≠ "How many people own their home"
Addendum: That almost 35% of all homes are owned by someone who doesn't live there shouldn't be a point of pride..
That is the percentage of homes owned by the people living in them, not the percentage of Americans who own homes.
Over a third (35%) of single family homes are owned by landlords, not by the people living in them. When a third of your single family homes are not owned by single families, then there is an issue.
The 26% number another person used might be percentage of US adults who own a home, but I can't find that number anywhere. The difference is most single family homes have 2 adults (or more) but is only "owned" by one of them. Or that is a statistic that uses multi-family housing where most of the families are renters.