I haven’t seen anyone really answer the why of it, which is that the industry developed using a floating glass tool called a hydrometer which measures the specific gravity, or density, of liquids.

When you boil the wort to prepare for fermentation, you end up with a sugary liquid that is denser than water or alcohol. Water has a specific gravity of one, and the specific gravity of the wort is increased by everything you dissolved into it. You would float a glass hydrometer in it and lets say you get a reading of 1.055.

After fermentation the yeast has converted much of the sugar to alcohol and decreased the specific gravity. You measure a second time, and multiply the difference by a constant factor to get ABV. let’s say after fermentation you got a reading of 1.015.

1.055 - 1.015 = 0.04
0.04 * 131.25 = 5.25% ABV

We label with ABV because that was how it was calculated, and remained the same regardless of the quantity served.

There is a similar process for distilling as well. Before these methods people didn’t know the exact amounts, which led to fun things like navy and admiral strength.

Edit: also the 131 figure really should vary based on temperature since it is derived from the ratios of the density of ethanol and water. The higher the ABV the more important it is to factor temperature, and distilling requires more sensitive measurements and tools. But for beer, using 131.25 is fine and has about 0.2% error up to around 10% ABV.