Stabilization feature not available in the Final Cut Pro X Inspector

Concerning your question "why", I guess that since angles can vary wildly, it may not make a lot of sense to try and stabilize the multicam as a whole. For example, one angle may be a handheld camera, another may be on a tripod, and yet another may be still images (I often do that with seminars, where I use an additional angle with png files of the slides).

Open the Angle Editor by double-clicking the multicam clip in the timeline, and apply stabilization to each of the angles separately.

It really should work the way you except. Why should a whole shot need to be analyzed for stabilization when only one clip has a bump in it?