I too have had this same problem just recently. I have Office 365 Pro Plus licensed by my university (C2R), and I purchased my own license of Visio Professional 2016 through Open Academic (special licensing for students, faculty, and universities). Apparently,
Open Academic licenses are volume licenses, not retail (like other student versions of software, or even the version offered by my university, which I would have imagined to be a volume license). Open Academic licenses are also MSI/Windows Installer-based,
whereas the university software is C2R, and for whatever reason software from both installers cannot coexist happily on your system.
That said, if your Visio license isn't a volume license, there's a workaround that I managed to sort of get working:
https://technet.microsoft.com/library/mt703272.aspx
I say sort of, because I managed to successfully install Visio Professional 2016 alongside Office Pro Plus 365, but even though the article is for installing volume licensed version of Visio 2016 alongside a retail license version of Office, the version
of Visio that was installed didn't accept a volume license for activation. However, if your Visio license is retail, you might be able to get it to work.
Note: I had to completely uninstall Office from my system before running the Office Deployment Tool. But as I said, when I ran the ODT, it successfully installed both Office Pro Plus 365 and Visio Professional 2016, although it didn't accept my volume license
for activation.
Good luck, Manjit! Let me know if you find success, because I'm still looking for success myself! >.<