Dan is an archaic honorific, akin to 'sir'. Cupid is an archaic god of love, the Roman rebranding of the Greek Eros. That's all.
The first uses of the phrase 'Dan Cupid' appear to be in John Grange's 1577 novel The Golden Aphroditis, in which he frequently referred to Cupid as such. Not too many years later -- 1598, or a bit earlier -- William Shakespeare penned Love's Labors Lost, in which he referenced:
This wimpled, whining, purblind, wayward boy,
This signor-junior, giant-dwarf, Dan Cupid,
Regent of love-rimes, lord of folded arms,
The anointed sovereign of sighs and groans,
Liege of all loiterers and malcontents,
Dread prince of plackets, king of codpieces,
Sole imperator and great general
Of trotting paritors -- O my little heart!
It is probably Shakespeare's use that solidified 'Dan Cupid' as a sleeper meme. Throughout the centuries, Dan Cupid has been a fairly well-used reference, either as a silly first name for a mostly-naked putti, or as a clever literary reference to show off your classical education -- and often, a bit of both.
Dan Cupid has appeared as a character in plays, and then a spew of movies and shorts in the early 1900s, a short story by Harry Houdini (Dan Cupid--Magician, 1907), an episode of Fat Albert (Fat Albert Meets Dan Cupid, 1975), as a reoccurring character in the Big Nate comic strip. Dan Cupid is also the unofficial mascot of Loveland, Colorado (Cupid, dressed up as a cowboy). But most often, he is simply a one-off character passing through a narrative, shooting off an arrow, and leaving the average reader with a slight sense that they've missed the joke.
BQ22