Lyric discussion by rjennar 

Cover art for Oh Gin lyrics by Velvet Underground, The

i figure this song's a play on words cause every time lou sings "gin" it comes off as sounding "jen", a common girl's name.

this way draws the parallels between wine and women.

or he could be referring to a girl named gin (my birth name is Virginia, but most people call me ginni or gin). it's not unheard of.