Unlink new IPhone from the old one

If you want to keep two iPhones independent of each other then best would be to have two different Apple IDs.

Otherwise, there are basically three types of sharing

1. Text Message and
2. Calls on the other device

iCloud Photos & Messages: iCloud now keeps your entire messages history updated and available on all your devices — even when you set up a new device. Learn how to keep all your messages in iCloud.

1. Set up text message forwarding
   1. On your iPhone, iPad, or iPod touch, go to Settings > Messages > Send & Receive. On your Mac, open Messages, choose Messages > Preferences, then click iMessage. Make sure that you're signed in to iMessage with the same Apple ID on all your devices.
   2. On your iPhone, go to Settings > Messages > Text Message Forwarding.*
   3. 
   4. Choose which devices can send and receive text messages from your iPhone. If you're not using two-factor authentication for your Apple ID, a verification code appears on each of your other devices: enter that code on your iPhone.
   5. 
2. To stop getting calls on one of your devices, just turn off the Calls from iPhone setting on that device. See the setup section above for details.
3. You can use iPhone Cellular Calls with any Mac, iPhone, iPad or iPod touch that meets the Continuity system requirements. It works when your devices are near each other and set up as follows:
   1. 1. Each device is signed in to iCloud with the same Apple ID.
      2. Each device is signed in to FaceTime with the same Apple ID.
      3. Each device has Wi-Fi turned on.
      4. Each device is connected to the same network using Wi-Fi or Ethernet.
      5. On your iPhone, go to Settings > Phone > Calls on Other Devices, then turn on Allow Calls on Other Devices.
      6. On your iPad or iPod touch, go to Settings > FaceTime, then turn on Calls from iPhone.
4. On your Mac, open the FaceTime app, then choose FaceTime > Preferences. Click Settings, then select Calls From iPhone.
5. If your carrier supports Wi-Fi calling on other devices, you can set up those devices to make and receive calls even when your iPhone isn't turned on or nearby. Learn about Wi-Fi calling.

To stop getting calls on one of your devices, just turn off the other devices on your iPhone settings.

Watch the GIF below