There's no need to save your windows key for windows 10.

Basically your windows 7 or 8.1 product key won't work to install windows 10 if you choose to fresh install. Once you have upgraded your system to windows 10 from prior version and shows activated, you can fresh install windows 10 over the SAME system without needing to input any key (just hit skip in the process when prompt for key).

Your system activation status is saved in MS server, that say, your system might not get activated when there's major hardware changes, no one knows exactly how do MS identify your system for windows 10 activation status, just yet.

Back to your question, no, the same key will not work in another system, unless it is a retail windows 10 key.