This has much to do with the crank-case scavenged valveless design of your typical two-stroke. Since it uses ports instead of valves there is a period of time where both the intake and exhaust ports are open and 20-30% of the unburned air-fuel charge goes right out the tail pipe. This allows for a very simple lightweight design but is no good for emissions or efficiency.
Secondary to that is that again since these small two-strokes have no valves they use tuned exhaust to try and retain the charge going straight through with a counter pulse, therefore it is very difficult if not impossible to do a catalytic converter in the exhaust as they will interfere with the tuning.
You notice the small 4-stroke is much better but not as good as the cars, this is because the 4-stroke has valves but no cat, it is also carbed instead of EFI.
2-strokes can be made with direct injection which delays injecting fuel until the exhaust port is closed by the piston, it adds much complexity and cost. Modern Evinrude E-TEC outboards marine engines have gone this route, whereas most other outboard makers have gone 4-stroke instead. Note that outboard marine engines do not require cats yet and emit about ten times the emissions per unit power that an automobile engine does. EPA is pushing for that soon though.
2-strokes can also be made with exhaust valves (called uniflow scavenging) this is usually done in large diesels since it also requires forced induction and no mixing of oil with fuel.
These small wankel like engines should probably have emissions similar to the small 4-stroke rather than the small 2-stroke as the Wankel design does not have intake and exhaust ports open at the same time but they are still carbed and no cat.