The only time I've ever (previously) heard of a "drinks cabinet" was when I was deliberately trying to search for noun adjunct expressions with a plural noun (on the notion that perhaps they existed somewhere, but I couldn't come up with any off the top of my head), and it came up as something said in the UK. And I've never heard of a "drink cabinet" till now (I take it from your comment that this might not be surprising; perhaps nobody anywhere uses it). In any case I'm not sure what this thing is. If I guess the meaning correctly-ish, an American equivalent might be "liquor cabinet," although I get the feeling "drinks cabinet" is a much more fixed, well-established concept.
I doubt too many Americans have ever uttered the phrase "beverages menu"; it's usually "drink menu" (I suppose I can imagine such a thing in a formal-enough establishment being labeled "beverage menu," but I can hardly imagine uttering the phrase otherwise). To me the plural noun adjunct thing is a strong indicator of non-American usage.
"Drink menu" is accepted, anyway, so I would assume they got "drinks menu," too.