Im going to have to agree that your device itself could be relevant. Heres why,
Its a google account you are speaking of, if you are using a tablet or phone, its possible you have a google app on your phone or used to, or its even possible the device is registered as a trusted device, that can receive security codes or texts..
Now, lets discuss your problem specifically..
When you sign in and enter the correct password ***** your google account, what exactly does it ask for next and how does it phrase it..
I ask because there are different security functions, that can sometimes be misunderstood for two step security and even if it is two step security I need to know exactly what its expecting from you next.
So after you sign in with the correct password, what exactly does it tell you next?