More to the point, when people use, "Google," as a verb, they mean to actually use Google, as opposed to using any brand of facial tissue available when saying, "Kleenex."
Exactly! You can't google something using Bing, for example. Not that you'd want to anyway. You can only google something using Google.
(Now I feel like I need to go wash my hands after mentioning Bing. Eww.)