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You do, in fact, own the house. It is encumbered in that if you sell the house, you have to pay back the mortgage first.

Buying a house doesn't negatively impact your net worth, even if you owe money to the bank, unless your home value nose-dives for some reason (eg recession/depression).

Here's an example:

Let's say today you have $100,000 cash in your bank account and no other assets to your name.

Your net worth is $100,000.

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Now let's say that you buy a house tomorrow for $500,000. You put 20% down ($100,000) and secure a loan from the bank for the final $400,000. Here's your new net worth:

Asset: House $500,000

Debt: Loan ($400,000)

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Total: $100,000

I wasn’t adding the value of the house into my napkin calculations. I was only looking at the debt and savings.

But you own a portion of the house and the bank owns a portion equal to the loan value.

The bank doesn't, really, they own your debt. The amount that you owe the bank does not increase or decrease with the value of your property.

Can you sell the house and keep the cash? You own the house, the deed is in your name. You have a debt with the bank. The bank has your house as collateral but the city chargers you taxes on it because you own it.

Yes you do!