Since Dionysus is the God of wine and drama, he can make wine appear with a swipe of his hand.
Nobody actually knows where he gets the wine, some people think he has the power to make wine really fast.
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Yes, the cult of Dionysus utilized a special jug that had two chambers. One would pour water into the jug and wine would be poured. Further consideration is that after Dionysus came Orpheus, and that these figures inspired the Jesus mythos.
They extracted Grape Juice or whatever type of fruit they were using and then bottled, and let the juice ferment, the same way we do it now pretty much.
Dionysus was a mythical being who would have stored wine in amphora, which are two-handled large earthen pots of a distinctive shape.