What's the voltage of a battery?
Voltage is indeed something that is measured at a particular point, but that doesn't mean its a property of only one point and in fact every point will have some voltage. In particular you can have a series of points that map out an "equipotential" where the potential is equal all along these points. A terminal of a battery is an equipotential so no matter where an electron "is" it will have the same potential energy U = e V.

Intuitively this is just like potential energy due to gravity. The battery is like a mountain with a flat top and level plane at the base. Everywhere along the top the altitude is equal and a ball will have the same potential energy. Similarly everywhere along the level base of the mountain the ball will have the same potential energy.