So something like "begging person" or "beggarly person"? I guess I can see where you're coming from. I've never heard people talking like that though, so it might not be as universal to western society as you think.
Personally, if you called me a German, a furry or a vegetarian, I wouldn't mind, even though none of these attributes encompass my entire existence. I guess the difference is that being a beggar carries a negative connotation, but I'm not sure that saying the same thing using slightly different phrasing really makes any appreciable difference.
It really doesn't though, in my opinion. If you talk about "someone asking for donations", I'd think of a volunteer collecting money for the local animal shelter. So if you actually wanted to communicate clearly, you'd have to go for something like "a person, typically a homeless one, who lives by asking for money or food". That's literally just the Oxford definition for "beggar" though. If you put that in the title of this question, it probably wouldn't even fit.
Firstly, I don't think that "shorty" is a good comparison, as that's an unambiguous (if mild) insult.
Secondly, it's not like anyone here is talking to any particular person calling them a beggar. If someone who was talking to me just called me "the German" instead of my name, yes, that might be a bit reductive and potentially rude. But if someone goes on Lemmy to ask "Why do Germans drive so fast on the Autobahn?", that's an entirely different thing. In that context it's simply a word that clearly conveys a meaning without having to use an entire sentence to explain it.