It seems you signed into your child's phone with your Google account and enabled syncing. Don't worry, it's an easy fix!
- On your kid's Bark Phone, open the ⚙ Settings app.
- Tap Accounts & Backup.
- Tap Manage Accounts.
- If you see your parent/guardian account(s) here, sign out.
- Then, follow this guide to sign in with your kid's Gmail:
Sign into Google Play Store or Google services