That's what "own" means with houses. You're a "homeowner" after closing, and you are the legal owner of the property, even though you still owe thousands to your bank.

That's my point. Debt is not an achievement.

I'd love to know their definition of "high-income family"

I think they mean paying a mortgage too. Paying off a house by 36 seems to require either a ton of money or a very, very cheap home.

One still owns a home when buying with a mortgage. The lender places a charge on the property while a balance is outstanding, therefore in addition to owning the property you also own the debt secured against it.