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Potential is different from potential difference.
In other words, bird body is like a resistor which is submited to the same potential in all of its points.
Therefore, the potential difference is zero between any point of the bird body which means no current and then no shock.
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Because birds only touch one wire at a time while it is hanging in the air (NOT CONNECTED TO GROUND), so the electric circuit is not complete. That's why they don't get electric shock.
