If your primer tells you what solvent it's based on, you can test a sample of material with the same solvent. One reason for not using a particular solvent with a particular plastic is that it dissolves the plastic. But some surface etching may be desirable to improve adhesion. Acetone, for example, will attack most of these, dissolving polystyrene pretty easily.
Instead, or before testing with solvents, a combination of density and the softening temperature should help. With reference to this table (sadly only in °F), we can see that thermoplastics differ in their density (a.k.a. specific gravity) and the temperature at which they can be formed hot.
For materials that sink, we can use the water displacement method to work out the volume of a sample, and weigh it, to calculate the density. Water has a density of 0.997g/cm³ (to all intents and purpose, 1g/cm³). You need reasonable precision for this sort of thing - curiosity like this is one reason why I have a cheap set of 0.1g kitchen scales. Acrylic is denser than polystyrene and ABS, but polycarbonate is denser than all 3. HDPE will float - just about.
The same table has a column for lower processing temperature. At this temperature, heated in an oven, it should be possible to bend a sheet of the material without breaking, and have it hold its shape. You'll need a good oven thermometer or infrared thermometer, as oven thermostats aren't accurate enough. Start cool and work hotter.
Unfortunately both ABS and PS soften at around 260°F (130°C). And they have the same density. Sheet polystyrene is rather brittle at a cool room temperature (CD jewel cases are made of PS); ABS much less so.
Thin polycarbonate bends in a springy way even down to very low temperatures. So does acrylic, but less so. Another table this time in °C, shows temperatures below which plastic become brittle. These are rather soft boundaries.
Acrylic is a bit harder to scratch than polycarbonate, but without comparing to known pieces it's hard to judge.
If you were going to compare to known materials, they all tend to have distinctive smells when you saw them or heat them (not to burning, but higher than the minimum forming temperature) - but you don't want to inhale great big lungfuls of the fumes anyway.
These are my guesses as to the most likely materials. All are hydrocarbons materials, and will burn with a yellow flame and give off a fair bit of soot. They can also have additives to make them less flammable.