Both phrases look very much alike, but "please" is not the literal translation
of "proszę". The phrase could be the polite answer to the waiter's question:
Q: "Can I get you anything (to drink)?" A: "Wino, proszę"/Proszę (cię) o wino
Wine, please/ Proszę (cię) o wino - Wine, please/ I am asking you for wine/
May I have some wine, please/ Some wine, please
It is not obvious here, that "Wino" is in accusative here, but it really has to be, because the verb "to ask for" goes with the accusative - Biernik (kogo? co?):
Since the noun "wino" ends with "o", it is a neuter noun, so the accusative is the same as the nominative - Mianownik (kto? co?). Non animate masculine nouns (ending with the consonant) behave the same way, the feminine ones change the ending from "a" to "ę", though:
Mleko/Piwo, proszę - Milk/Beer, please (neuter nouns)
Chleb/Ser, proszę - Bread/Cheese, please (non animate masculine nouns)
Kawę/Herbatę/Kanapkę, proszę - Coffee/Tea/Sandwich, please (feminine)
These are polite, easy, and popular: May I ask for wine, please? -
Woman or Man - Mogę prosić o wino?/ Czy mogę prosić o wino?
These are a little bit more sophisticated, but difficult to pronounce:
Woman - Mogłabym prosić o wino?/ Czy mogłabym prosić o wino?
Man - Mógłbym prosić o wino?/ Czy mógłbym prosić o wino?