Let me first note that the voltage itself does not have any physical effect - the harm comes from the electric current. This happens in two ways:
- via the Joule heat when high current runs through the body
- via triggering the heart defibrillation when the frequency of the alternate current is in resonance with the heart frequency, that is 50-60 Hz
Let us now look at the parallel circuit formed by the bird and the piece of the wire between its legs. The voltage, i.e. the potential difference between the bird's legs is definitely not the same as the potential difference between the wire and the ground (which is known to be a few kV). The current in the wire (a few Amperes) is partitioned between the bird and the piece of wire between its legs: $$i = i_{bird} + i_{wire}.$$ The potential difference between the bird's legs is $$V = i_{bird}R_{bird} = i_{wire}R_{wire}.$$ Solving these three equations we obtain: \begin{array} ii_{bird} = \frac{iR_{wire}}{R_{wire} + R_{bird}},\\ i_{wire} = \frac{iR_{bird}}{R_{wire} + R_{bird}},\\ V = \frac{iR_{wire}R_{bird}}{R_{wire} + R_{bird}}. \end{array} The resistance of a human body ranges from 1000 to 100000 Ohms, depending on whether it is wet or not - this could be a good estimate for the bird. The resistance of a copper wire is a few Ohms per thousand feet (depending on the wire diameter). That is the piece between the bird's legs has resistance of a few mOhms. Thus, $$\frac{i_{bird}}{i} = \frac{R_{wire}}{R_{wire} + R_{bird}} \approx \frac{R_{wire}}{R_{bird}} \ll 10^{-6},$$ i.e., the current flowing through the bird is a millionth part of the current in the wire or even smaller! It is too minuscule to cause any real damage. In other words: the piece of the wire between the bird's legs short-circuits the bird.