A Get Menu person sent me a link to save their contact card. What does that store about me?

Nothing, unless you choose to fill in the short form on it — a name, a number and an optional note, so a follow-up is not from a stranger. Leave it blank and the card still saves to your phone perfectly. Nothing is read off your device, no contacts are touched, and it is not connected to any restaurant account. If you allowed notifications, that is a permission you can withdraw in your phone or browser settings whenever you like.