1 stick of butter equalling half a cup or 8 tablespoons is ubiquitous to the point that recipes refer to measurements by half a stick or two sticks instead of 1 cup of butter and we all understand what measurement it means.
How did the size of a stick of butter become so uniform?
If this doesn't find any experts in /r/AskHistorians I'll try /r/AskCulinary next, but I'd prefer a historians answer.
Butter is sold by weight (1 pound is the usual denomination), not volume (cups), although now we cook with butter by volume in America, absolutely. If you look at older recipes they will rarely give butter in measures other than weight. Sometimes if you do not buy fancy butter it comes in that big block that is just a pound and you have to cut up yourself, wrapped sticks are a convenience. The stick is sold as 1/4 a pound (4 oz) therefore, and also happens to be half a cup in volume. The cup is derived from the gallon, it's 1/16 of a gallon. The density (and therefore weight) of butter matching up roughly (not exactly!) to cups is a bit of a happy cooking coincidence, the units are not otherwise linked.
The Wikipedia article on butter gives the first instance of stick-wrapped butter as 1907, but cites a thesis from 1948 for that, so take that with a little salted butter. The dubious but fun Google Ngrams doesn't show "stick of butter" as a common phrase until the 50s and 60s.