I have my Bic Camera and Yodobashi camera point cards on my phone so I don't have to carry the cards around any more. I know those are connected with my personal data but, in both cases, the discount is more like 10% so it's worth it to me (and even if you carry the card, it's still tracked). I don't register for supermarket cards but yeah... I have a ton of stamp cards for restaurants and bars, most of which I keep in a separate case in my bag. There's only a few I use frequently enough to carry in my wallet. And stamp cards aren't tracked like point cards. The discount seems to vary wildly from place to place.
BTW, I also have Edy and Suica set-up on my phone, which I think is pretty convenient, even though I assume someone is using that data to track what I buy and where I go.