Symbolism, Imagery, Allegory
No—it's not a play on the word "horny." But it could very well have been—Shakespeare is all about the innuendo. "Horns" in Shakespeare's time symbolized that your wife was cheating on you; you were a cuckold.
Horns show up consistently in the play as a symbol of marriage (and the idea of a husband being whipped in marriage). Horns are an interesting symbol, because while they’re used in this context to denote a whipped guy, they’re also a thing that wild beasts have.
Don Pedro jokes about the "savage bull" with horns at the beginning of the play when Benedick refuses to marry (and again when Benedick has chosen to marry after all):
BENEDICK The savage bull may; but if ever the sensible Benedick bear it, pluck off the bull's horns and set them in my forehead, and let me be vilely painted, and in such great letters as they write "Here is good horse to hire," let them signify under my sign "Here you may see Benedick the married man." (1.1.257-262)
The inevitable horns that come with marriage (remember Beatrice jokes that if God sent her a man, he’d have to send horns too?) represents marriage as a process by which wild animals are tamed.
Horns are joked about throughout the play, but over the course of the action they become less fearsome as the characters come around to viewing horns as an ornament of love. At the end of the play, Claudio promises that Benedick will have horns, but they’ll be gold-tipped, like Zeus’s horns were during his lusty conquest of Europa.
Horns are a thing that wild animals have, but as the characters learn to accept and be excited about marriage, they come to symbolize the sexytimes of marriage.