Irritating Chrome logo

That is because you are running the app from the desktop, not where it should be. It should not be in the downloads folder either. First you need to quit the app (command+Q). Then get uninstall instructions from their website. Once it is uninstalled, download ot again. This time do not run it until it is properly installed.

The icon will not move because that is the actual app. If you move it, it will stop working.

Looks to me like it could be the disk image of your Google Chrome app. If it is in fact a disk image, if you right click it (or Control-click it) you'll see an option to Eject the disk. Assuming Chrome isn't running, go ahead and eject it. Then try dragging it to the trash.

That is the install drive that you are ejecting (deleting). It does not impact the actual Chrome app in your applications folder. You do not need this one on the desktop

I took the liberty of posting your photograph in the correct orientation:

Select (single-click) that icon, then select File > Open. Describe what you see, or better yet, post a screenshot.

To learn how to capture, edit (if necessary), and post a screenshot to this site read the Appendix in Writing an effective Apple Support Communities Question - Apple Community.

Mac tips for Windows switchers - Apple Support

That seems to be the actual application icon. It should be in your Applications folder and will normally move there once you install an app. Running apps from the finder is not the best way, so drag it into the folder.The icon in the dock is a shortcut.

I believe that is the logo that is generated when first installing Google Chrome. Just drag it to the trash. You should also have another Google Chrome logo in your Apps folder that is the real applicationj