That must be new because it want there a few months ago.
It appears when you get the first alert - click to silence it, and choose "indefinitely." That option only appears for Family devices.
Turning off the alarms would be a great start (they already added that?). What I really want is shared tags though, i.e that my wife can search our house keys and not just me. They are our house keys and everything else is extremely integrated in iOS so it feels weird to not be able to share the AirTag.