Unless something big changed since I left two years ago, the org chart is totally visible to the entire company, and so are levels except when someone explicitly opts out (most do not bother).
Pretty uncommon at tech companies, but it's entirely based on the company's policy. The company can fire you at any time (as long as it's not one of several specific bad reasons like racial discrimination), but it's the company that employs you, not your manager. So only the company can fire you.
Generally big tech companies don't give your manager the authority to blindly fire you, but only because that's a decision the company made.
Yeah, in corporate world, at least you're paid to not care, and can change jobs easily. In academia, you're paid shit, and changing labs is not nearly as easy.
What I tried, in a similar situation, was being clear that I did not support the company or its leadership and for them to never contact me again. It worked.