Nicotine gum surely does have the same teeth-cleaning function as regular gum. And, now that I look it up, it seems there's basically nothing online that claims nicotine gum itself causes staining (results are mostly just a study or two that establish that nicotine gum helps remove staining from prior smoking, which yeah that's probably true). I, or someone else talking to me, might've gotten confused about a claim originally made about smokeless tobacco products and thought it applied to gum too.

Nicotine gum isn’t “chewed” like regular gum (unless you have a huge tolerance for nicotine or want a tummy ache) —- usually you just chew it intermittently to release more nicotine and then keep it in your mouth. So while it won’t cause damage to the mouth, I can’t imagine it will have the same effect as chewing Orbitz