Means more than that, it's an invitation to actually drink "several other beers". The proposed translation is wrong, this is a countable, not uncountable use of it. The latter you said, on the other hand, is a good possible translation (e.g. -A me piacciono le bionde e le rosse. -Ah, solo quelle? Non bevi qualche altra birra?) although it's a bit far-fetched (would be more natural with piacere than with bere).
I think "do you drink a few more beers" is the best translation, because "altro/a/i/e" means "other" but also has a nuance of "more". So when you ask for more pasta, you'll be asking "altra pasta, per favore!". But what's really important is to remember that "qualche" is used with singular but means plural. So "qualche giorno fa [=ago]" = "a few days ago", "ho letto qualche libro" = "I've read some books" etc. Hope it helped :)
Do I understand well that this sentence may suggest different types of beer, like "do you drink a lager, then a pils, and continue with a porter and a good ale?" i don't know whether Italian has such nuissances for beer because (as far as I experienced) drinking beer in Italy is like seeing Napples and not dying... or drinking grappa in Ireland :D
This sentence can be intrepreted in so many different ways: because I thought that the question implied that it could mean whether the "you" is drinking some other beer than the person asking the question is drinking, as in a different brand. I know the sentence isn't meant like that, but please DuoLingo, change the translation for this sentence to something like: "do you (want to) drink some more beers?" because this is really confusing.