Two people should definitely be able to log in to the same machine (though not at once) - it’s how we have our test machines set-up in our office.
You actually don’t need to do a re-install to set up a new account: https://support.zwift.com/hc/en-us/articles/205139326-Can-I-create-another-user-on-the-same-install-
That being said, as long as you log in with your own information (email address and password), you can log in to *any* machine Zwift is installed on since all your profile data is stored on our server rather than locally.
Quick question: are you logging in to the game and then going to zwift.com? Or logging in separately to each? Because the website and the game each require their own log-in.