...is it a standardized measure?
(Out of curiosity, is it a regulated measure? Would I be prosecuted for selling a stick of butter that doesn't weigh 1/4 of a pound?)
In my country we wouldn't describe a solid in cups, just grams.
Because you ran into it in the first place. You think cookbooks are giving out subjective instructions?
"You said stick, but I don't know what that means" is a pretty reasonable conclusion to draw when you encounter the usage. "You said stick; you must not believe that words have meanings" is something else again.
An obvious analogy for the lack of precision of "stick of butter" outside of US culture is my culture's "block of butter" - typically, a block of butter in my country is 500g. But not always.
So how do you know how much butter I intended you to use if I say "add half a block of butter" in my biscuit recipe?
How big is a block of butter where you live? Do they even sell it by the block? Or is it just sticks? Rough mental calculation, there's about 4 and nearly a half sticks to the block.
I guess there's a reason humans developed and use units of measurement.
Anecdote time! When I was young I tried to follow a recipe off the Internet that specified a stick of butter. So I diligently cut a long skinny bit of butter from my block of butter, a little confused as to what difference the shape made. The biscuits were rather awful.
There are some exceptions like Amish butter that come in a 2 pound roll (round block). Kerrygold and Finlandia butter from Europe adapt to the 8oz and 16oz standards just like a US company would adapt to UK or likely EU standards.
This is probably one of my country's fanciest butter makers: https://lewisroadcreamery.co.nz/collections/butter
Cue the funny anecdote. We've been using au pairs to help with the kids. It took about 6 months for one of us to notice that when a recipe called for "1 cup" of something, one au pair would go into the cabinet and grab a random glass and use that to measure. Our glasses range from 4 oz. to 18oz, so there were tons of size disparities. We didn't notice for so long because she was usually making rice, and the ratio was the most important part, and the ratios was cups to cups.
To be clear, that disparity had just never occurred to us, and in no way did we blame her. All of us (including her) had a good laugh over it and we showed her where the measuring cups were kept.
I'm American. I lived in the US until around 8.5 years ago and I moved to Norway. I've had to look up conversions many times so that I could cook comfort food.
Other folks have answered the regulated measure stuff, but FWIW: Every butter maker sold sticks where I was from (Indiana) and all the sticks were the same size. Margarine brands did the same so long as they weren't sold in tubs.