I didn’t say it “didn’t” … I said it’s “not supposed to. “
“ … These alerts are triggered only when an AirTag is separated from its owner … “
Ref: AirTag - Apple
That said, this forum is full of posts citing examples of airtags not functioning quite as “advertised.”
My own tags don’t perform as they are “supposed to” either, although thankfully I don’t experience the same “alerting” issue as you; which indeed would be quite annoying.
So … is your wife’s phone in-range of her tag when it’s alerting YOU ?
Is your wife’s iPhone signed-in w/ the same AppleID used to “pair” the tags? (i.e. Can she see the tags on HER phone within FindMy?)