Hi @Yuuu,
You may be clicking on the desktop shortcut pointing to a specific profile, and there may be a profile problem. You can directly go to the executable path of Microsoft Edge (normally it's C:\Program Files (x86)\Microsoft\Edge\Application\msedge.exe) and test if double click on msedge.exe works.
If it works, you can consider deleting the old desktop shortcut and then creating a new one. Same approach applies to Chrome as well.
EDIT
Please also check if you have enabled "Run this program as an administrator" option. You can right click on msedge.exe --> Properties --> Compatibility to check this option. If you have this option enabled, please disable it. Screenshot as follows:
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Best Regards,
Shijie Li