The stuff you are talking about is Expanded PVC (trade name Sintra). It can be cut and screwed together just like MDF...so yes you could build a cage with it....but as Joe points out you really need a heat press to get the most benefit from using it. Sintra is flexable and brittle so if you cut and screw it together it is more likely to bend and break apart. By heating, bending and sealing it you don't create moving joints that can break apart which keeps it more structurely sound.
So depending on the benefit you were hoping to get will depending on if you should use it. I have built cages from it and also MDF and outside of some added weight there isn't much difference in the end results.