Not possible. It has something to do with how Chromium/Chrome identifies its windows/class. (Unlike Firefox, Chrome/Chromium ignores StartupWMClass). The --class flag is also ignored in modern versions, so while it won’t produce an error, it is no longer honored.
I tried every approach you can imagine, and it never succeeds.
I even tried this guide in the past, but no such luck. Perhaps you will find success with it? (The author of that guide even admits it doesn’t actually work for Chrome; it only works for Firefox.)
If all instances of Chrome are closed, the next profile you run will become the dominant icon in the Task Bar. All subsequent instances, even for different profiles, will merge with the first icon.
EDIT: If anyone can figure this out and demonstrate that it works, I will be beyond delighted!
EDIT 2: I’m referring to Linux (and more specifically KDE). I know this works under Windows 10 (and I currently do what @zauberschloss desires for KDE, except on Windows 10 instead.)