Short answer: We've no idea. Schulz never told us.
But he did play around with the concept in several strips that ran over an extended period of time, in which Snoopy wondered if his little friend might be a variety of different birds, and Woodstock attempted to imitate each.
It started with a crow (May 11, 1979), and then, more than a year later, Snoopy gets serious on September 13, 1980, by checking his "Guide to Birds." He first suggests that Woodstock might be an American bittern, and several strips follow during which dog and bird buddy imitate the noises of various species: a Caroline wren (September 22, 1980), a rufous-sided towhee (September 24, 1980), a yellow-billed cuckoo (September 29, 1980), a Canada goose (October 1,1980), a warbler (October 3, 1980) and a mourning warbler (October 9, 1980). Snoopy finally gives up trying to figure it out, and hurts Woodstock's feelings by saying, "For all I know, you're a duck" (October 11, 1980). Snoopy takes it back with a quick hug, at which point it becomes clear that it doesn't matter what type of bird Woodstock is; the only important fact is that he's Snoopy's best friend.
https://en.wikipedia.org/wiki/Peanuts
