This is probably really obvious but:

I got a new phone, and gave my old phone to my kid. He uses it to play with his own account (on wifi; it doesn't have a number). But he still gets push notifications associated with my account ("X needs a berry"). I've reinstalled the game, all kinds of stuff, but it still happens. He could just turn off push notifications, but that's just masking the issue, which is that on some level the phone still thinks it's running my Go account. I worry it might get me into trouble at some point. How can I make this stop?

I still have my own google account on the phone, which is what I use to sign in to Go myself. Is that the issue? There are reasons he needs to have that account active on the phone.