Yes, you could as well say "wir trinken einen Wein / ein Bier / einen Schnaps ...", and in a restaurant you could order "ein Wasser" or "einen Saft".
All these drinks come in glasses or bottles that you could count. When there is no need to mention the kind of container and the exact quantity in it (because it doesn't matter, is clear from the context or fixed on the menu) you can omit it and simply say the number. Usually, and in particular in a situation like the one you are describing, you say "ein", because that's the number you are starting with. It doesn't necessarily mean that it cannot become more ;-)