Does this sentence carry the same suggested meaning as in English, that the speaker is referring to drinking alcohol unless the context specifies otherwise?
No. It means in the same context of "I drink when I am thirsty." So it could be water, tea, coconut water, juice, etc. It could mean alcohol if the person is an alcoholic :-)
Thanks, I've come across quite a few euphamisms and other such things in my learning to date. So I've concluded that it is important to check when I can.