What if you run TestComplete.exe at a standard Windows command prompt and not in Powershell? Does that make a difference?
Honestly... I don't know. It just seemed like something to try, in case there's something specifically in the PowerShell environment that triggers opening anything as a project.
I've reproduced the issue on my side.
Try use "Start-Process TestComplete.exe". Also if you need to use some parameters read help info for PowerShell "Start-Process" command.