I was going to try the new element.io web app. Something went wrong when I tried to verify my session.

I'm using Firefox on Fedora 32. I removed element.io cookies, and tried again, but I still have this problem.
This is likely one of your browser extensions blocking requests to matrix.org.
I confirm that disabling Privacy Badger made element.io working.
What is going on ?
What changed in riot, now element, that my pricacy badger breaks it ?
Privacy Badger has a heuristic which says if a domain is accessed from multiple sites, such as matrix.org would be (from riot.im and from element.io) then it assumes it is a bad domain and blocks it.
I just installed the stand alone version and got exactly the same error...
@perremba Today I discovered that the standalone versions are somehow connected to the web versions. There was an option in the web version to install the standalone application. When I did so from a web version that did not work, the standalone version had the same issue.
In my case, installing from the Brave browser resulted in a problem, however, installing from Chrome went fine. Here is a screenshot showing that the standalone application somehow knows where it was installed from.

I don't think you're installing the standalone version but in fact the browser PWA version, which is no different than using your browser.
The standalone application should be installed from https://element.io/get-started
@t3chguy What is PWA?
You are right, that's what I installed.
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I just installed the stand alone version and got exactly the same error...