"Polyurethanes are the only class of polymers that display thermoplastic, elastomeric, and thermoset behavior depending on their chemical and morphological makeup."This is nearly an entirely true statement. Polyurethanes can show a vast array of mechanical behaviors .
Surprisingly, this versatility arises only from a fundamental weakness in polymeric nomenclature. Addition polymers such as polyethylene, polypropylene and poly(vinyl chloride) are named based on the narrow set of monomers used to make the polymer and therefore the properties of resulting polymers are equally narrow. Contrast this with polyurethanes, polyureas and other condensation polymers that are named on the linkages that form during the polymerization. Since we are naming the polymers only on bonds that form from very small segments of the monomers, the balance of the monomer is free to be practically anything and therefore the polymers' properties can be too.
Knowing this then allows us to expose the weakness in the statement that ONLY polyurethanes show this vast array of properties. Start by looking at other condensation polymers. First, there is the kissing cousins of polyurethanes, polyureas, which can be just as versatile. Polyesters would be another class of materials with a vast array of properties. While polyethylene terephthalate (PET) gets the lion's share of the attention when the word "polyester" is mentioned, unsaturated polyesters are used as adhesives and thermosets and behave quite differently than PET. Thermoplastic copolyesters also exhibit highly elastomeric properties.
I could go on with other examples, but I also want to return to the previous statements about addition polymers. Consider acrylics. While these are addition polymers, they also vary just as vastly as polyurethanes in their properties, but that is a result of three factors. First, there is the richness of the chemistry that can be done to modify acrylic acid giving a huge array of monomers that all fall under the heading of "acrylic". Secondly, the polymerization chemistry allows for the extensive use of comonomers. Lastly, and probably most importantly, it is very easy to polymerize acrylics - it can be done at room temperature with minimal amounts of equipment. Contrast this with polyolefins which are made only at very large scale with industrial equipment at high pressures and temperatures using air-sensitive catalysts. If you want a special copolymer of ethylene, hexene and decene, you better be prepared to order several million pounds of it. If you want a special copolymer of methyl methacrylate, 2-ethylhexyl acrylate and acrylamide, you can make it yourself this afternoon. The same is true with many polyurethanes - you can easily mix them up in your lab, no special equipment needed.
Which then raises the question of what is the ultimate range of any given class of polymers? In many cases, we are limited to what is commercially available even if greater versatility is theoretically possible, and then limits our thinking about what that class of polymers is like, even if large. Regardless, polyurethanes are not the ONLY class of polymers to display thermoplastic, elastomeric and thermoset behaviors. Not at all.
 I've posted in the past that whenever someone tosses me a sample and asks "what's this polymer?", polyurethane is never a wrong answer - you can probably make a polyurethane that would come close to mimicking that sample.
 Remember Newton's Third Law of Isocyanate Chemistry - "For every isocyanate/polyol reaction, there is an equal but faster isocyanate/polyamine reaction"