There is no way to open the Files screen in a new tab from within a design file in the browser version of Figma.
When you click the “Back to files” menu item in the main menu from a design file, you are forced to leave the current page. Middle clicking or right clicking doesn’t change anything.
The current behaviour forces me to go through a loading screen when I open the design file again. I also loose the state of where I was.
Navigating between different screens is what links are made for. The same problem happens in the Files screen when you want to open a design file in a new tab: yes you can write click and select “Open in a new tab”, but links offer this functionality natively. It is not possible to middle click on files to open them in a new tab, which would be more natural.