After a short time, they ask you to buy them a drink.

I have never encountered or heard of this behaviour. I would be rather startled if someone I had just met asked me to buy them a drink. I'd guess they were too poor to get their own (and with all respect to poor people, my interest in pursuing a relationship with them would substantially diminish).

I can understand your explanation, but I would find an opposite explanation just as plausible (they are trying to determine if the cost of a drink is a mere trifle to you, hence buying them one = good).

Is this a culturally specific thing? Where is this action, with this meaning, a standard pattern of behaviour?