Mice will eat anything and everything. When I first bought this house many moons ago I found out mice were eating the bar of Dove soap in the bathroom. They loved the stuff and I took to baiting traps with it and got rid of them. I did not immediately figure out what the marks on the soap were.
I also bought a box of poison but never opened it, just put it under the sink. The mice nibbled through it and ate the stuff
I do not know if the stereotype of mice eating cheese extends to other cultures but it is also deeply ingrained in Spanish culture and there are several sayings. To say someone was fooled or took the bait there are several sayings like “it was given to him with cheese”. Preparing a trap you would say “setting it with cheese” etc. So the idea cheese = trap comes directly from the idea that mice like cheese.
It could be this stereotype goes back many hundreds of years and spread all over Europe. I wonder if other European cultures share it…