I'm trying to understand this better, could this also mean "What are you drinking?"? Is it also the same as "What would you like to drink?", is it simply the context that would differentiate these?
Absolutely, all your proposals are correct, in different contexts.
- at breakfast, what do you drink? = au petit déjeuner, qu'est-ce que tu bois ? (habit)
- what are you drinking right now? = qu'est-ce que tu es en train de boire, maintenant ? (continuous)
- drinks all around! what do you drink? = tournée générale ! qu'est-ce que tu bois ? (invite)