I have a Papermate "Titanium" mechanical pencil. It's about a $4 pencil with a white eraser if that helps guess the chemistry. It is roughly 15-18 years old and has been used in various temperatures and also sat for long periods in a drawer during it's time since purchase. I probably have not used the eraser in maybe 2 years, but I definitely used it, including erasing, 2 years (or so) ago.

I went to erase something yesterday and it was kind of sticking to the paper. On looking at it, getting curious, and squeezing it, I found that the eraser had become like a thick bubble gum. It actually has the consistency of a kneaded (charcoal art) eraser, but not the functionality. This is an eraser that worked like a good wooden pencil eraser... Never was too squishy before.

Obviously this is a curiosity question... I can drop in a new eraser or just buy a new mechanical pencil. But what about the chemistry of erasers would cause one to work well for 13+/- years, and then turn into a gummy putty over the course of two more?

It would seem that a lot of erasers are a rubber-type matrix with an abrasive embedded in it. Ozone, sunlight  and general oxidation from air can break that matrix down that gives it rigidity. Think of old tyres with the cracks in the side walls; that's the matrix breaking down. In your case, it looks like it broke down into a mush rather than into bits.