Most browsers on MacOS will advertise the currently open tab via Handover when the browser is in focus (or at least this is the behaviour from a user perspective), which allows a user to open the tab from another device.
Currently, Orion on desktop will act as a handler for incoming Handoff messages (i.e. those from another computer or phone) when it is the default browser, but won't advertise visited tabs/sites over handoff for other devices.
(linked - https://orionfeedback.org/d/873-handoff-current-tab-between-macos-and-ios, reopening as Vlad asked for a new issue as last one was closed)