You control your data in B-DAY. Here is exactly how.
Contacts
Contacts import is optional and read-only. To disable it: iOS Settings → B-DAY → Contacts → off. You can still add people manually.
Notifications
In the app: Settings → Reminders → turn off Birthday reminders (and Daily summary). System-wide: iOS Settings → Notifications → B-DAY.
iCloud Sync
iCloud Sync is off by default. Turn it on or off in Settings → Backup/iCloud → iCloud Sync. With sync off, your data stays only on your device.
Delete your data
Delete a single person from their detail screen, or use Settings → Backup → "Delete local data" (this device only) or "Delete local and iCloud data" (also removes it from your private iCloud, when signed in to iCloud and online). Uninstalling removes on-device data but not iCloud data.
Exported backups
If you exported a JSON backup, that file is wherever you saved it. Delete it like any other file to remove that copy.
Contact
Privacy questions or requests: alexblessingx@gmail.com.