1) Give me the borshch means give me that specific borshch. There could be many kinds of borshch there, or "the borshch" could refer to "the borshch you cooked".
2) This sentence is not necessarily an order at a restaurat. Actually, it's not. I wouldn't tell a waiter "дайте...", I would say "я би хотіла..." -> I would like. This sounds like someone talking to their family or friends.
I'm a native American English speaker. Even for me, it is hard to definitively say when the "zero article" can be used. For Borsht, I believe this falls under a mass noun. Quoting from https://www.thoughtco.com/zero-article-grammar-1692619, "Mass nouns are those that cannot be counted, such as air or sadness. They also include nouns that are not usually counted but that can be counted in some situations, such as water or meat.". It is common to say "I'll have soup", "I'm making soup" or "I'm making a pot of soup". We would not say "I'll have Borsht", but that is only because we don't have Borsht as an option.
Somehow "I'm making soup" and "I'll have soup" sound OK to me, but "He gave me soup" or "Please, give me soup" sound weird... The first two sound like they describe some general concepts, e.g. "making soup" is an activity, "I'll have soup" sounds like you're ordering from a menu and "soup" is like the name of an item. But "He gave me soup" sounds too literal and specific (to me) to have it without "some" or "the"... I'm confused :')
Don't compare "borshch" to concepts like "advice" and "money". That doesn't make any sense. Your comparison to "soup" was a good example. However, you said "He gave me soup" or "I found soup in the fridge" sounds OK to you, and I still feel that it sounds strange and should be "He gave me some soup" and "I found some soup"...
I do know you're a native speaker and I'm not. This is not about you convincing me :D It's about the number of people who comment about it :) One person is not enough of a statistics.
But I think now there have been enough native speakers saying it's OK and I'll add it x)
In the early exercises 4 versions appear: borshch/borsch/borscht/borsht
However, in the later ones "borscht" disappears. I wonder whether I should add it there, or remove it from the early ones :) I don't know what the idea is: maybe "borscht" is not common and it's been decided to remove it...
It's not accepted because the idea that we had before was, there needs to be something before "borshch" e.g. "the borshch" or "some borshch". However, there was a comment above saying no article is needed and "Give me borshch" is correct as well. However, to me it still sounds weird like "Give me soup" or "Give me potatoes" - just sounds off.
I'm not a native speaker though. So I would like if more native speakers comment on this first. I'll also try to find some source online explaining it, just couldn't find anything yet :D