Landlords, banks and agencies routinely ask for a copy of your ID. Marking it is a small step that meaningfully limits what that copy can be used for later.
An unmarked photograph of your passport is a general purpose credential. The same image marked with a specific purpose and date is obviously tied to one transaction, and is far harder to present convincingly to anyone else.
Placement matters more than opacity. A mark in the corner is cropped off in seconds. A mark running across the document cannot be removed without damaging the thing it is attached to, which is the entire point.
This is exactly the kind of file that should never be uploaded to a stranger's server to be edited. Handing your passport to an unknown service in order to protect your passport is a poor trade, and it is the reason this runs on your own device.
Most do, and many expect it. If someone insists on an unmarked copy, it is reasonable to ask why.
The purpose and the date. Specific text is far more useful than the word COPY.
No. Nothing leaves your device, which matters more here than almost anywhere else on this site.