[dragonwriter]

> With two-factor authentication you are happily providing gmail with your phone number.

Which I also provide to Google because all of my phone numbers are forwarding numbers for my GVoice account, so that's not a big deal.

> They say they need this to send you a verification code when you log into your gmail account.

Sure.

> Well, if that's the trick, they don't need your phone nr at all, they can do ip and os check anyways..

How can they determine its valid without the second factor the first time you log on from a particular device? That's a key feature of 2FA (plus, if you ever use a shared computer, you don't want to choose the option to never ask for a code again on that computer!)