Why can't it be "He drinks the first cup of wine"?
cup = xícara
Could this be translated also as "he drinks his first glass of wine" or it does not have that meaning?
It keeps the same meaning, although it is not strictly literal.
Why cant I say "He is drinking" his first glass?
= ele está bebendo
As I understand it...the present tense is represented in "he drinks", and "he is drinking".....The way you have it says it specifically, he is drinking....
Only in specific contexts you can use "Present Continuous" as "Simple Present", but it should be avoided. You can be more accurate by using literal translations in this case.
he drinks his first glass of wine?