This would seem to be something that would be local - to the point of differing from bar to bar.
I've probably seen more references to "sending the kid for a bucket" in tales about frontier towns in the US than in European sources. (Of course, I've seen more US references than Euro?)
These sources frequently imply that it is a practice associated with certain "ethnic populations," but the Irish are not significantly more often cited than several others.
A suggestion seen somewhere - with the US's Calvinist(?) traditions, you had to bring it home if mama wanted some, since "ladies" couldn't be seen in a place that served it. If papa got it, people would think he was drinking too much, but if you sent the kid, everybody knew it was for the whole family.
Of course nowadays it would be for the kid - so that's illegal most places.
Most places in the US, a "shot" is the amount of liquor added to other ingredients in a "drink" served in a bar. This doesn't mean a lot though. Most places, a shot means whatever the management thinks they can get by with, and can vary from a half-ounce (or less) up to about an ounce and a half (the usual maximum that's common), except in Nevada where by law a shot is 2 ounces. (or is that just a folk legend?).
And isn't the old joke: Q: How do you tell when an Irish family is on vacation? A: They go to a different bar.
Surely it's because things are different at the other bar?