Colly is an old word for black, so any common black bird could be meant, as Lighter suggests. Several other meanings for colly, but none is pertinent to the old rhyme
Some have suggested that the pear tree is from the French word perdrix, which can mean partridge, but since the original English toy-book speaks of "a partridge in a pear tree," I see no need to make this assumption. A simpler explanation is that the composer of the game needed something to fill out a line and happened to pick a pear tree on which to perch the partridge.

According to Eckenstein, in northern France, the game is called "Les dons de l'an," Gifts of the Year. The gifts start out with one partridge (no tree), followed by two turtle doves, three wood pigeons, four ducks flying, five rabbits trotting(?), six hares a-field, seven hounds running, eight shorn sheep, nine horned oxen, ten good turkeys, eleven good hams, twelve small cheeses.
I don't know which is the older of the two games.