Hi, I looked into this yesterday and was able to successfully change the icon. It was complicated and I'm not sure if there will be any adverse affects to changing the folder permissions (which is required). If you want to try this, I suggest reading through everything that I did before you even start. I also used the same info from the superuser.com page.
First I changed the folder permission so I could edit the files then I replaced the "icon" image (actually a .png file) with a file of my own.
I downloaded and installed the "TakeOwn.reg" file into my registry.
#2 on this page:
https://www.tenforums.com/tutorials/3841-add-take-ownership-context-menu-windows-10-a.html Also read #5 through #8.
I opened the following folder by using Start / Run and pasting in the folder name:
C:\Windows\SystemApps\Microsoft.MicrosoftEdge_8wekyb3d8bbwe\
I right clicked on the Assets folder and selected Take Ownership to change the folder permissions.
I then opened the Assets folder where I was presented with 100's of files, most PNGs that are probably used throughout the Edge software. You will see many copies of the default blue icon of various sizes. I'm not sure how many you would actually need to replace but I tested several until I found the one that is used as the taskbar icon. For me, the filename was: MicrosoftEdgeSquare44x44.targetsize-24_altform-unplated.png
I replaced it with a .png file of my own choice.
I reopened Edge and the icon change was immediate.
Lastly, I went ahead and removed the "TakeOwn" from my PC.
#4 on this page:
https://www.tenforums.com/tutorials/3841-add-take-ownership-context-menu-windows-10-a.html Also read #5 through #8.
There's 2 more questions that I have and I can't answer for you.
#1. Will changing the folder permission cause problems? I don't know.
#2. Will it always be that same file for everyone? I don't know. You might need to replace a different file, maybe multiple files. It may depend on your taskbar size.
Also there is a big Windows Update which is the process of being rolled out to everyone now. That big update may actually reset the icon back to normal. Keep that in mind.