Office 365, Office Upgrade Installation
Removal of old Office versions may not be necessary to install the latest version of Office from the My Software page on Office 365 - especially if the previous version was an install from Office 365 - however it is generally best practice to remove all previous versions of Office before a new install.
Upgrading Office software - does it require removal of old Office software?
Additionally, you may need to Deactive the previous version on the Manage Installs page of the Office 365 site before beginning a new install. You can find the Manage Installs page by clicking on the name in the top right hand corner, clicking on View Account, and the Manage Installs link is in the Install Status box.