Staudey Yeah, it seems to be pretty similar. I'm playing around with the Budgie code to come up with a good solution here.
Edit to add: Using the same trick as with VirtualBox works, but it means that the profile chooser window doesn't get an icon, so it's really just shifting the problem.
Edit 2: For some reason, the icon isn't updating correctly when the first window is removed and the browser window opens. Even though it doesn't have the running visual, clicking the icon minimizes/maximizes the Chrome window correctly. So it's at least correctly figuring out that it's a part of the same application group, and that it is running. I don't know if this is the same as the old VirtualBox issue.