What could possibly cause a "melting" reaction between vinyl and plastic?
It is probably a consequence of the plasticiser not the polymer

It is hard to be sure what has happened here without checking the details of which polymers are involved which the picture can't make clear. But we can make some intelligent guesses based on commonly used polymers.

First: the cable isn't made from natural rubber. Most likely it is made from plasticised PVC. This is a good and cheap cable material with good insulating properties and doesn't decay like rubber from interacting with atmospheric oxygen. This involves a lot of plasticiser, usually something like a phthalate. The cable might contain 10-40% of plasticiser by weight. 

Depending on the exact polymer formulation and the specific plasticiser, the plasticiser can migrate into other materials if they are in contact for long periods of time. When I used to work in a lab, for example, the cheap PVC tuning we used in low pressure gas lines would sometimes accumulate a thick liquid in the lower parts of the tubing loops if used for years: this liquid was plasticiser that had migrated out of the PVC over years of use. This migration is worse when other volatiles are in contact with the polymer.

This may be what is happening in this case. In warm conditions the plasticiser in the PVC might migrate and damage other plastics. It isn't possible to tell from the picture what the breadboards are made from but ABS is most likely in a good quality breadboard, though acrylic, polystyrene or polycarbonate are all possible. All of these polymers can be affected by the migration of plasticiser from PVC as this study shows (though it depends on the specific plasticiser). Good choice of materials for the board and plasticiser can minimise this problem but cheap breadboards and cables are likely to have worse problems.

In short, this is the likely problem here.