To turn your Caller ID on/off on an iPhone:
Open Settings
Scroll down to Phone and tap on it
Tap on Show My Caller ID
Toggle the switch to show or hide your Caller ID
To turn your Caller ID on/off on an Android device:
Open your Phone app
Tap on the three dots in the right hand corner
Tap Settings
Tap on Calls (or Supplementary Settings if using a Samsung device). If you don't see one of these options, look for a More settings or Other Call Settings option
If you see an Additional Settings option please tap on it, if you don't have this option you can skip this step
Tap on Show Caller ID or Caller ID (Please note that this option may have a slightly different title depending on your device)
Tap on Hide Number or Never (This option may also have a slightly different title depending on your device)