It's the current which matters.

The bird has a capacitance to the rest of the world which depends on
its size. The current which flows up the bird's legs is the voltage on
the antenna divided by the capacitive reactance of the bird's body.
For small birds the capacitance is only a few picofarads and so the
current is extremely small. The bird continues to sing its happy
song.

Whereas, a human body not only has a much larger capacitance it may
also complete the circuit by having the other hand in contact with
grounded metal work. Or the victim might be standing in his bare feet.
The current due to capacitance of the human body alone is enough to
cause burns. The pain is sufficient to stop the human from singing
and begin using bad language.
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Reg.