"I will let it drink" makes sense also. I think if using "he/she" the translation should be "Saya akan membiarkan dia minum." Thus shows if it is a person.
Yes, it makes sense, as it can be an animal drinking.
Can it mean drinking alcohol in Indonesian?
That works too, but usually people refer to drinking alcohol as "minum-minum" and "minum" as drinking in general
Generally, Indonesians don't drink alcohol. So 'membiarkannya minum' is always meant 'drink water or other water based beverages'.