Light fixture measuring 120V for both hot and neutral
That's not neutral

Normally, wires cannot be re-tasked.  But there is an exception in Code which allows white wires to be used as hot.  The white wire must be marked with tape or paint.  Prior to that, marking wasn't required "if the usage is obvious".  

This is most often seen in switch loops, which is the usage here.  

On both sides, black is always hot.  One side's white is neutral back to supply. The other side's white is switched hot returning from the switch.  We don't know which is which, but also, don't really need to. If in doubt, connect the lamp to one black/white pair. If the lamp lights, that pair is supply and that white is true neutral.    

Lamps connect to switched hot, and neutral.