The Atlantic Canary Islands are named after Serinus canaria - The Canary bird. Actually the canary bird is named after the Canary Islands which get their name from the Latin 'Insularia Canaria' which means 'Islands of Dogs'.
Yes, the wild canary comes from the Canary and surrounding islands. The domestic canary is a descendant of the wild bird. However, the bird is named for the islands, not the other way around. The Canary Islands were named for dogs, real or otherwise, the Latin name Canariae.