Reddit user Leopeva64 spotted the change tucked away in the latest developer build of Canary. If you use Canary, you can enable Send Tab to Self 2.0 by typing or pasting chrome://flags/#send-tab-to-self-v2 into the bar at the top, then setting it to Enabled. This changes the way tabs are shared across platforms, replacing system notifications with a small icon that will appear next to the address bar.
As Android Police points out, the move away from notifications is far less intrusive as it takes system pop-ups out of the process entirely. It also works independently of your system notification settings, so even if notifications are disabled for Chrome the new sharing function will still work. Additionally, this prevents the shared tab from being hidden under other notifications that may appear—keeping the information set to the side and safe until you decide to pull it up.