Just as in Firefox and Chrome, Microsoft Edge too has an option of private browsing which would not store cookies, history and temporary files.
However this option is not directly available from the + sign used to open new tabs and also there are no shortcut keys for doing this.
To launch private browsing in Edge :
Click on the 3 dots icon next to the address bar and choose New InPrivate window :
This will enable private browsing mode. The default InPrivate tab will open the search window with a brief description about what this mode does.
Tabs opened from now on in this window will delete browsing history, cookies and temporary files once they are closed.
Happy private browsing.