Why is it called a fifth of liquor?

Answer 1

Since a "fifth" of liquor has less liquid than a quart, than a "fifth" must be 1/5 of a gallon, right?

Originally there were two types of "gallon" in use for liquid:

The "Wine Gallon" or Queen Anne's gallon of 128 oz or

The "Water Gallon of 160 fluid ounces, or 10 lbs of water. AKA "Imperial Gallon"

The US Gallon of 128 ounces was ratified later based on the Wine Gallon.

one fifth of that is a US bottle of liquor

The fifth, which used to be 757ml, has now been reduced to 750ml for convenience with the rest of the world which uses the metric system.