When I open my existing solution in Visual Studio 2012 professional, I got the above error. This used to work but something messed up after the computer is rebuilt.
Soluton for me was to run the standalone TFS installation. This is for 2017, but worked to fix my VS2019.
https://visualstudio.microsoft.com/thank-you-downloading-visual-studio/?sku=TeamExplorer&rel=15
Bit of background...
I'd been having this issue since 2017. Tried all the usual suggestions on here, cache clearing, config file editing etc. but would always have to come off my organisation's network to check code in, as it was proxy related. TFS was trying to connect to a proxy server ip that had been decommissioned, no one from the service desk could work out why. Couldn't find any reference to this old ip in any config files.
Installed VS2019 hoping it would solve the issue but it persisted. Eventually I thought of trying to uninstall it as a component and re-installing it. Couldn't find a way to do that but in my search I found the standalone installer for Team Explorer, albeit for VS2017.
After installing, I ran this, opened up Manage Connections in the Team Explorer connections page and did what I needed to connect to the repository from there. Worked first time. Closed Team Explorer 2017, opened my VS2019, and everything seemed fine in there as well.
Amazed my problem is finally solved!