I suppose you know what the plastic family is, for instance polyoxomethylene and polyoxoethylene.
One excellent method is to measure the density then. On a cylinder or a brick, you measure the dimensions with 10-3 accuracy at least, scales give you the mass at least as precisely. Since POM-H and POM-C differ by several %, you get a definite answer. Be careful with humidity absorption in PVC-PVA and in PA.
If you only have one object of this polymer, you might try to adjust a liquid mixture until the object just sinks in it. I haven't done it by myself; check for bubbles.
In the case of POM, you could pyrolyse it to determine what proportion of C makes the initial mass: big difference between -C-O- and -C-C-O-. Less huge with polystyrene copolymers and with polyamides, bad with polyolefins.
Then you have the low-tech methods, which require reference samples. Burn chips, compare the smells, the fumes, the flames and droplets. Or scratch the surface with a needle, feel the movement with your fingers. These are rather mechanics workshop methods, probably not what people expect from a chemistry lab.