Greensleeves seems to be only one of a plethora of ballads (laments, what have you) about love, faithless love, etc., some of which which appear in "A Handful of Pleasant Delights, 1584, by Clement Robinson and divers others" (a 1924 transcription by Hyder E. Rollins and published by Harvard University Press, reprint Dover 1965).
Another is titled "A Proper Sonnet of an Unkind Damsel to her Faithful Lover," which seems to follow the same line.
Several are reproduced, including Greensleeves, at: Ballads
It seems that this kind of song was popular at the time. We try to attribute special meanings to "Greensleeves" but they end up being "Just So" stories.