It used to use Omaha. They recently rewrote it and Chrome now uses an installer framework fully maintained in their own tree. It's conceptually similar but cleaned up.
And since the Chrome fork Edge is part of Windows now, this updater already runs on all Windows machines (e.g. the MicrosoftEdgeUpdateTaskMachineUA scheduled task).
They did not even use Windows update for that one.
Can you force Edge to go through Windows Update? They have GPOs for download, but I didn’t see anything like that in there.