Transferral of ssh client key from M$

I believe as long as they have accounts on the linux box they should receive a new key. I don't think SSH bases it's exchange of keys off of user names. From my experience it seems to have based it off of the different machines I've been logging in from. I can log onto my linux server at home from any computer as root and do not need any keys, because they are generated for each new connection. I'm no expert on this, but have never needed to copy my keys from my Windows pc to my linux box in order to login to another linux box. I hope this makes sense.

I know it is possible to generate new user keys on a Linux client, but my whole point is that I want to use the same client key everywhere, both from Linux clients, and from the Windows client...