When you buy a license we assigned it to your Google account, it means you can access from any computer never the less you are connected to your Google account. If after receiving email confirmation of license assignment you don't have access to the app it means you are not well connected.
99% of the case are due to usage of the function in Gmail "Add account" (check support link)
To quickly check you have access to the app you can open an incognito window, link. Shortcut : Ctrl + Maj + N (Windows, Linux et Chrome OS) ou sur ⌘ – Maj – N (Mac)
Connect to your account and follow the link of the web app.
Best way to use multiple Google account on one computer is to use the Chrome session.
This will create different instance of Chrome for each of your profiles. By this way you will have the session for your Gmail account and the session for your Google Apps account. No issue with cookies etc... because everything is separated.
Create chrome profile : support link or copy paste this link in the address bar : chrome://settings/createProfile
How to ? >