Trying to install Adobe FrameMaker 2022 but it gets stuck at below screen.

You're not logged in. Log in at https://account.adobe.com/ using the Adobe ID you used to download the trial, and try again.

It's definitely trying to "phone home" to get you to sign in with your Adobe ID. Are you working in a corporate environment? If so, check with your IT to see if they aen't blocking some of the ports that Adobe uses to call out.
Did you try the link in the "If your browser is not responding" message?

Hi,

If I would have your issue, I would probably feel lost ;).

And I do not have an idea as well.

I would contact the Adobe support. Here are your support contacts:

https://helpx.adobe.com/contact/enterprise-support.other.html#framemaker

Good luck.

Best regards, Winfried

FM is likely using whatever is the default browser on your machine (which might not be Edge). Is there anything unusual about it?