That's kinda how danes pronounce it if they are just talking casually. But it should be more clear I agree.
It can in fact be both, it depends on context. In this sentence, I think someone has mentioned the thing that's being drunk before (For example 'Ser du saften?' - 'Ja, katten drikker den'). Generally, 'det' would be used rather than 'den', but both are gramatically correct.
my guess is that the thing we're talking about (which we don't know in this sentence because of lack of context) is common gender. Den/Det follow the gender of the object.