I have a single machine pro license, I want to know If I can use the same key on the same machine, with different user accounts?
Is the single machine license valid for all user accounts on this PC or linked to a single user account?
The sign in on the Editor is for usage with the Unity Services, mostly.
You can answer your own question by creating a new free account and finding out what happens if you sign in with that free account.
You may not be allowed to sign in. If you are allowed to sign in with a free account but you still have access to Pro features then the answer to this question is ‘No’.
If Pro features are not available then the answer is Yes as it would be apparent that the License keys are being managed. These days with Unity 5 (I could be wrong) the Pro features are extra services and do not change the editor as the full feature set is available for both.
Of course this just explains workings of the Editor. Your license is very much tied to your account. Remember you had to go through the process of Activating your license.
You should really contact the support/licensing teams through the website contact form. I’m just spouting conjecture.