Why makes ground "ground"?
The bird doesn't have a voltage of 0V, its voltage is floating, which means that it has the same voltage as anything it touches. 

As soon as the bird touches ground, the bird gets a voltage of 0V. When it takes off, it is floating again. When it touches the powerline, it gets a voltage of 120V. 

Only when it touches the powerline and the ground at the same time, one end of the bird has 120V, and the other end of the bird has 0V, which means that a current will flow, killing the bird.

Also, what makes ground ground is convention, nothing more. If the powerline was DC (unlikely I know, but for simplicity, bear with me), we could declare that the ground has a voltage of 1,000,000V, and the powerline has 1,000,120V. Or that the ground has -120V and the powerline has 0V. But neither of those would be very practical in daily use.