The purpose of characterizing some cases of eating as eating out is to mark them as departures from the default: eating at home. When someone says ‘Let’s eat out tonight’, that conveys something like ‘Let’s eat out tonight, instead of at home, as we usually do’. Saying of someone ‘He eats out frequently’, implies that he is different from average people, who eat most of their meals at home. Trying to transpose this to drinking would produce awkward results, because one’s home is not generally regarded as the default location for drinking (i.e. drinking alcoholic beverages, separately from a meal), in the way in which it is regarded as the default location for eating. That’s why, it does not normally occur to people to say ‘drink out’, and any attempt to introduce such a phrase could not make it analogous to ‘eat out’ (under the existing social conventions).