Yes, the sleeve the piston travels in. That's much better than some engines that run the piston in an aluminum cylinder.
Most generators like this run at a constant 3600 RPM's because that is what's needed to produce 60 Hz AC power. The engine carburetor is set up to run at the governed 3600 RPM's and vary the throttle opening as the load changes. Some, better, higher priced generators run at 1800 and art geared to spin the generator at 3600. Inverter generators, on the other hand, vary the speed of the engine as the load varies; that's why those small Hondas are so whisper quiet. They use DC alternators and invert the power to the 120 VAC. They run much quieter and use less fuel. My Yamaha is not as quiet as the Hondas but didn't cost as much; still much more than all these others, though.
I used to have a 4 KW standard noisy type. For use at a campsite it was hard to bear; I would not use it unless we were alone, what the FS calls dispersed camping (in the trees someplace). If I was using something like that at the cabin perhaps a three walled "enclosure" to deflect sound away from the cabin could help a little. You have to watch the air flow though.
I did improve the exhaust noise a lot with a super duper silencer, but that did not help out all the mechanical noise. An engine designed specifically for quiet generator use also has more webs cast into the internal and external structure. The engineers do analysis to prevent mechanical harmonics increasing the mechanical sounds. Engines used in most "garden variety" generators are not generally designed like that. But everything that is done to make it quieter boosts the price.