@Mikkel Witzell, Thanks for posting in Q&A.
To clarify this issue, please check the following.
1.Can you view saved passwords before switching wo Web Sign-in? Could you please unassign the web sign-in policy and check if we can view password in Edge and Chrome?
2.When it prompts for a "device password" is there other choices that you can select and enter, such as PIN, smart card?
Here are links you can try to fix the issue.
https://www.technipages.com/fix-microsoft-edge-does-not-remember-passwords/
https://www.guidingtech.com/top-ways-to-fix-microsoft-edge-not-saving-passwords/
Non-official, just for reference.
Please check above information, if there is any update, feel free to let me know.
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