How to refer to the piece of furniture that bars use to store their drinks?

Use "shelves". That's the word that would come naturally to me.

I thought that was called a bar. When you see an "antique bar" sold, it is often the piece that houses the liquor, not the counter where the patrons sit.

The patrons sit at the bar.

The barman, or barback, grabs a bottle off the bar back.

The bar itself can have shelving and different "stations" for draft beer, ice, refrigerators of alcoholic drinks, topping (limes, little umbrellas, etc.)

The built into the wall bar behind the barback is a hutch. Atop sits the bar back display of a variety of drinks. Top shelf is the more expensive bottles, well is the go to (cheap) versions.

Those things with the upside down bottles?

Optics.

Otherwise, we just say shelf.

Edit: from working in hospitality in Scotland.