Pages you've added are recognized in any browser
The extension checks your Screenly account for the page you're on, so a page added from Chrome shows "Update Asset" in Firefox or on another computer.
You added the sales dashboard from Chrome at your desk. A week later you open it in Firefox on a laptop to change how it's saved, and the extension knows it's already on Screenly.
When you open the pop-up, the extension asks your Screenly account whether the page you're on is already there. If it is, the button reads "Update Asset" and "Save Authentication" shows how that content was saved. It's the same answer in Chrome and Firefox, on any computer, and after someone else has signed in to the extension in the same browser.