Hi Provolone_Wolf,
Thank you for using the Apple Support Communities! I understand your Google Chrome icon is not appearing on your application. I'd like to help.
It sounds like Google Chrome is working OK, but the icon that represents this application is incorrect. You mentioned you deleted and reinstalled Google Chrome. To confirm, did you follow the steps as suggested by Google Support here: Uninstall Google Chrome - Computer - Google Chrome Help. If you did not, try these steps to uninstall, and then reinstall Google Chrome to test for the issue once more: Download & install Google Chrome - Computer - Google Chrome Help.
Otherwise, it sounds like you took some great steps already. Were you able to reproduce the same issue while in safe mode, or did the icon appear correctly?
I would also like you to test a new user account to isolate if this is a user-specific issue, or a system-wide issue. Here’s how: How to test an issue in another user account on your Mac - Apple Support. Based on the results in the new user, you could continue to the section titled 'If the issue doesn't happen in the other account' or 'If the issue happens in the other account' to continue troubleshooting.
Let me know how this goes. Regards.