I mean, in American English I don't think it's that common to hear "buy candies". Typically it's a singular plural (just "buy me cookies and candy"), unless you specifically say "various types of candies", or even just "various candies".
How do Britons and Australians say it?
Hah, that's funny - as a Brit, a lolly to me is a very specific type of sweet. Basically, some kind of sweet or candy that you suck on, with a wooden or compressed paper stick to hold it with. Often, but not always, a boiled-sweet type. Chupa Chups would be a classic example. If it's fruit juice or similar frozen into a shape with a wooden stick to hold it with, it's an ice lolly. Calling a gummi bear a lolly is very strange to me! I had no idea of that usage.
iOS actually has an emoji of a lolly, I have no idea if it'll work here on DL, but I'm sticking it at the end of this message to see...
In New Zealand (and, I'm guessing, Australia), a lolly is the general word for any confectionary – not frozen, though, that's an iceblock – or you can use sweet (to sound British) or candy (to sound American). Candy is most definitely uncountable though, to me. You can have lots of candy, or just one "piece of candy". I don't know where on Earth the word "candies" is used, but it bothers me every time Duolingo says it.
@TheFinkie "Candy" can be countable or uncountable in the U.S. "Countable candy" is a strict subset of candy: round, hard things you suck on, no stick. Of course "candy" also has the normal "uncountable plural" meaning "types of candy." "Конфеты" isn't actually a completely general term for "candy" either to the best of my understanding, and it doesn't overlap well with "countable candy," so I would say your objection to "candies" would be on pretty solid ground even in the U.S.
I think to translate the feeling "some", it would have to be печенья here, particularly since конфет is also in the genitive. I suppose one could argue from the Russian, if you heard печенье и конфет it could mean "cookies and some candies", but in this context it makes sense that both are partitive.
You can use genitive as a means of demonstrating partiality (in this case "some") in some cases.
Another example: Хочешь чаю (or also somewhat acceptable - хочешь чая) would mean "Do you want some tea?" (more common perhaps in a home or restaurant environment where you're asking for a cup of tea) as opposed to just "Хочешь чай?" (Do you want tea, which might be more common in a store where you're buying bags).
Someone can correct me on this, but I don't think in today's modern world it really makes a difference either way and you'd be understood (obviously if you tell someone to купить тебе конфеты they're not going to buy Aisle 11 at the supermarket).
Because печенье is a mass noun in Russian, and is treated as singular, just as in English one would say "Buy me some sugar", not (usually) "Buy me some sugars".
(I can just about stretch it to say if someone wanted more than one type of sugar, they might say the latter, but it is a stretch).
It's just one of those little foibles of the language :) Potatoes are also a mass noun in Russia, but hair(s) is (are) not. So instead of eating a cookie while you brush your hair, in Russian you eat cookie while you brush your hairs.
COOKIE is the name of small confectionery products from non-yeast dough / Печение - это название мелких кондитерских изделий из бездрожжевого теста/ There are different types of this dough /Существуют различные виды такого теста/ Butter dough. It has a lot of oil / Масляное тесто. В нём много масла/ Sugar dough. It has a lot of sugar. /Сахарное тесто. В нём много сахара/ Bisque dough. It has a lot of eggs and sugar/Бисквитное тесто. В нём много яиц и сахара/ Shortbread dough. There is no sand in it. It crumbles like sand./Песочное тесто. В нём нет песка :D Оно рассыпается, как песок/