In my company setting up the office email on our personal phone is optional. But if we choose to do that, they install this entirely new 'profile' with work apps and controlling software on that in the name of security. I chose not to install it, but some would surely do it. Probably no case can be made against them as they don't force their employees to install mail on their phones.

I did that for convenience (though now I removed it), but the work profile seemed to be well compartmentalized from the personal part, and there was a strong emphasis on this fact before the installation process. If they were to wipe it, only the work profile would be affected. I fortunately never had to try it...

Right, but if you add your account to their organization so that you can get onto their network dont be surprised when they push policies.

No but if you add their MDM to your personal PC they can push policy to it.