Wow, I never knew this. In fact, the problem appears to be even worse than you said: it doesn't just clobber the stabilization for the current clip, it actually copies the stabilization from the applied clip! (So if the clip you're applying from doesn't have stabilization, it appears to delete it.)
So the bottom line is that the Stabilizer data is available through the Stabilizer tab (second one) in the Tracker tab and it seems to be global in nature. It's not attached to any particular Correction node, including the default one.
You can delete all nodes, and the Stabilizer data disappears, but then reappears when you add even a default Correction node back again. But if you Apply a grade, it disappears. Or more specifically, the Stabilization for the that's being Applied replaces the existing Stabilization.
I can't see that it makes sense for Stabilization to ever be copied/applied. (I guess if you could Destabilize it could be useful for a full-bodies compositing capabity in Color, but...)
Unfortunately, the same thing will happen if you do Stabilization in the Color page. Applying another grade will clobber your stabilization anyhow. Even if you're disciplined and always Stabilize after you're "done" with color grading, all it takes is a last-minute tweak and you've got no stabilization or even the wrong stabilization.
So I think they need to make a decision: a) never copy/apply Stabilization, b) have a "Protect Stabilization" button in Stabilization -- defaults to On -- that doesn't allow incoming overwrite of Stabilization, or c) make Stabilization a node-specific effect just like other Tracking. I'm thinking A or B make the most sense.
