Apple - How to stop this icon from appearing on the taskbar?

That's the "handoff" feature. You can turn it off at: Settings > General > Handoff.

When you turn it off, that device can't send or receive handoffs. A little bit more info.

You could also use a different iCloud account on your phone/iPad if you wanted handoff to work between a Mac and one device but not both iOS devices, but more than one iCloud is a hassle, so disabling handoff entirely is much cleaner.