System on external drive using on iMac and MacBook Pro
It's because authorizations are OS or profile based, but machine bound. Every time you start the OS on a Mac that is different from the last time it started up, all authorizations become invalid and have to be reissued and are stored on your disk again for the machine that you currently use.

Look in to Mobile Homes to see if that would work.

By the way, what you are doing is probably totally unsupported, so bugs are to be expected, even if it works.