If I have a home mortgage and have only paid off 20% of the principal, then the bank technically "owns" the remaining 80% of the home. So, why isn't the bank required to pay 80% of the property tax every year?
You have a misunderstanding of what a mortgage means.
The bank does not own 80% of your home. Instead they have lent you some amount of money, and they have a lien which means they can possess your home if you fail to pay it back. They are not responsible for any costs of your home.
I know people popularly say "the bank owns 80% of my home" but it is not true in any technical or financial sense.
(There are some exceptions for unusual kinds of 'mortgages')
Slight addendum to the already posted correct answer:
The institution who lent you the money has no reason to pay the property tax on your building. They aren't using water, taking showers, flushing toilets, getting police and fire protection, driving on city streets, etc...
However, they do have a profound interest in ensuring that you pay the property tax.
As explained, if you should default on your payments, the mortgage holder can take the house, sell it, and take what is owed from the proceeds.
EDITED to address comments:
But, if you have run up an unpaid property tax bill before defaulting, the local authority gets first crack at the value in the property to pay the tax bill, ahead of the mortgage company. This in turn reduces the amount available to the mortgage company to satisfy the mortgage in default.
No lender likes to see their available collateral diminish in value, so they will either demand an annual receipt showing taxes paid, or add the property tax to the mortgage payments you are making.
For the same reason, one often needs to supply the mortgage company with proof of adequate fire/flood/tornado/earthquake/whatever insurance...
And if one borrows more money with the property as collateral, the terms first and second mortgage denote not chronological order, but rather the order in which the lenders, upon default, can get their balance owed from the sale (until it runs out).
When I sold my home (Toronto, Canada) I took back a mortgage from the purchaser, that, among other things, allowed me to inspect the mortgaged property (with reasonable notice) to ensure that the property, that I no longer owned, was being maintained so as to preserve its value.
As long as you make your mortgage payments, the bank has zero property rights in your house.
The bank cannot evict you from the house.
People from the bank cannot enter the house without your permission. They certainly cannot just decide to move in one day.
The bank has no say in what furniture you put in the house, what color you paint it, or what sort of trees and flowers you plant.
Yes, if you fail to pay the mortgage, the bank can bring legal action against you to force you to pay, including ultimately the right to sell the house to collect what you owe them.
But then, if you take your car to an auto mechanic and then refuse to pay the bill, the mechanic has a similar right to sell your car to pay the bill. Does taking your car to an auto mechanic mean that now the mechanic owns the car? No, not unless you don't pay the bill.
Suppose you buy a toaster with a credit card. Who owns the toaster, you or the credit card company? You do, of course. Even though you haven't paid for it until you pay the credit card bill.
I had surgery a few months ago and the doctor put stents in my heart. I haven't paid most of my share of the bill yet. Does the hospital own my heart? No. (My girlfriend does. :-) )