"I need beer."

The difference is (in most of the cases):

Мне нужно + noun - i need something (i'm lacking something) - Мне нужно пиво (молоко, хлеб)

Мне надо + verb - I need to do something - Мне надо сходить (принести, выпить)

"Нужно" can substitute "надо" but usually not vice versa.

I think is sentence sounds fine, especially if you think of it as something to say to a friend. Certainly, as a college student, I've heard it often enough.

In English, if I said this in a store people would think I'm an alcoholic.

Same in Russian. It may be a normal phrase in the context though:

• "I need beer for the upcoming party"
• "Мне нужно пиво для предстоящей вечеринки"

In Russian there is no difference either you want a bottle or you have a permanent need for such a sentence. But maybe the creators meant an alcoholic? Who knows.

this unit is titled, "shopping." So I think it's somebody saying that beer is on the list, just like milk or eggs or whatever.