But if the asset bubble bursts and your home loses most worth.
Luckily the primary function of a home is to provide shelter. You don't have to worry about being out on the street if you're unemployed (pending you can afford local taxes).
It's possible that the house price falls by 10%. It is very, very unlikely that rents will crash by 10%; they're sticky, at best you'll see a very slow slide over years.
If you're still paying your mortgage, do you really own your home?
You own your home in this scenario just as much as the entity on the other side of the equation owns the income stream from your mortgage.
A mortgage is a collateralized loan. You agree to give up ownership of your collateral if you fail to meet the terms of you contract. So yes, you own the home now, subject to constraints.
Ownership is almost always subject to some constraints. Nobody says “do you own your home even though you can be taken out of it (and jailed) if you refuse to pay taxes on it?” even though that’s arguably a more extreme condition upon your ownership. While I am sympathetic to conversation about the broader meaning of ownership, this “mortgage => !ownership” thing is a tired meme.