Yes, the air cleaner mount is shaking because your engine is shaking, obviously.
Because of the design of the bracket and mount the air cleaner vibrates much more harshly than the engine itself; it seems like the vibration is actually amplified in to the air cleaner housing.
Here is a video of how little my '85 OM617 shakes at idle. The air cleaner doesn't move (good bracket and brand new mounts) but you can feel it vibrating quite harshly if you put your hand on it.
Smooth idle diesel 1985 300D W123 - YouTube
Now if your engine itself vibrates more than this then your air cleaner will be even worse off. You'll want to make sure you have good engine and tranny mounts, well-adjusted valves, etc. This will lessen the stress on your air cleaner bracket.
You can also do some to help absorb vibration. I think people have stuffed some thick rubber, like maybe some coolant hose, between the mount and in intake mainfold to help reduce vibration.