Tell us the make and model of the breakers. That may indicate that the panel was swapped after the house was initially built. The condition is dangerous for multiple reasons. When I have found this problem, it often was that two circuits went to a wall box and got tied together. In cases like this both circuits feeding that fixture(s) must be on the same pole of the service in order not to create a dead short circuit. Find and discontinue the interconnection.

What are you measuring voltage with?

The panel has been replaced, about 20 years ago. I'm not using anything to check voltage - literally the light stays on unless you switch off both breakers. I won't be doing any work on this myself - I am just trying to figure out how big of a mess I've got on my hands and how urgently it needs to be addressed.