I feel like it would be useful to know exactly how much alcohol is in a can or a bottle. Also why is alcohol the only thing measured in percentages and not sugar or caffeine or medicine?
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.
That still doesn't give us a reason why customers have this special calculated value. You could just multiply the volume by this percentage to get the absolute value. Why is the percentage preferred over the volume? I see no reason for either side.
Like I said, because the percent doesn't change with the volume served. If you are an 1800s brewer you can calculate the ABV from samples, and subsequently sell kegs of various sizes, bottles, which in turn can be served in various amounts and the percent doesn't change. And the industry never changed, nor the laws written. So it's the way it is because that is how they used to do it and how laws were written and there hadn't been a motivation for people to change that.
We do lots of things as a society because we've always done them that way, or it's good enough, and not enough reason to go through the effort of changing everything including legal language, etc.
Happily, in this case I think ABV is about the best way we could have inherited, maybe only second to alcohol by weight in terms of consistency across temperature.
Also, it's practical. If you know something is twice as concentrated (12% instead of 6%) you know to drink it more carefully, rather than if you get a jug of something and it just says how much alcohol is in there, then you have to mentally calibrate how strong it is by considering the volume of the jug vs. how much alcohol there is.
You have to print a separate label for each bottle size then. Much easier to print something like '10% strength' and slap the same label on all of your barrels and bottles regardless of their volume.
You still have to show total volume on each bottle. So the label is different anyway. Plus, it could have been done as they do to show singer content - amount per serving size and servings per container.
Present age does nearly the same thing though. 20% simply means 20ml in serving size of 100ml.