Customizing a program's taskbar icon (for all instances)

I want to customize the icon of a certain program (in this case Firefox). There are many instructions on how to supposedly achieve this by e.g. simply going to any shortcut (in Explorer or Desktop), Properties, Change Icon. Then pinning this shortcut to taskbar puts the customized icon there. But as soon as you click that icon, it loads a new instance and the new instance has the "real" icon. And the customized icon just sits there taking up space. I have no idea why Microsoft has intended it like that, it makes no sense to me.

Is there some way to change the icon so that all instances of the program show the customized icon?

I tried Resource Hacker. I remember I've been successful with that many years ago with some program. However, for Firefox, there is some problem. It seems that all programs have a different set of icons. Like 16, 256, 16.8M color icons, different sizes etc. For Firefox, I'm not sure if I'm creating exactly the right kind of .ico group icon. Whatever the reason is, if I replace the icons with Resource Hacker, Firefox won't launch anymore (nothing happens).

xTL, that was an awesome video. However, your Windows seems to work differently than mine. I'm running Windows 10 Pro 1803 (build 17134.286). When I follow that exactly, using VLC as an example, once I'm at the point where you double click the icon while it's in the TaskBar folder. Nope, my taskbar icon is NOT the one I changed it to with the Change Icon (as in your example). It shows the original VLC icon in taskbar (while in the TaskBar folder the shortcut's icon is the custom one). So, unfortunately that didn't work out for me.

However, meanwhile I was able to find out why Resource Hacker didn't work for me. I was doing it with Portable Firefox by PortableApps, this one: Mozilla Firefox, Portable (browser) | PortableApps.com and I didn't realize that its initial .exe is actually loading App/Firefox64/firefox.exe. Once I found out that, I could simply open that firefox.exe in Resource Hacker and then in its Icon Group I could ctrl+r replace 1 and 32512 with an (group) .ico file that I created in Online ICO converter (where I just uploaded a 256x256 32-bit PNG). And options on that site: [x] 16, [x] 32, [x] 48, [x] 256 pixels and (x) 32 bits. Then, after saving the file in Resource Hacker, it's done. Firefox has the new icon in taskbar. (Only in the jump list it shows the old icon for some reason, but that doesn't matter at all.)

As soon as the firefox.exe updates, I have to do the icon hack again. But this is great, I can now have two different Firefoxes running simultaneously, independent of eachother. The other one also has different coloring on the top of the browser for further distinction. The coloring is done by an extension called Envify. (However the color updates only after either first cltr+n or ctrl+t, but it's still quite neat.)