Apple id and devices

It sounds like you are sharing an Apple ID. If you don't want to continue to do so,

See How to create a new Apple ID https://support.apple.com/en-us/HT204316 See the data implications below.

Some of these things will apply to your situation even if you are not changing your Apple ID: What to do after you change your Apple ID or password https://support.apple.com/en-us/HT204071

 Keep in mind that some things available under a first Apple ID will not be available under a new Apple ID. For example, you won’t be able to transfer anything purchased from the App and iTunes Stores to the new ID. And, there are other issues to be addressed.

See what Eric Root say on this matter here:  hthttps://discussions.apple.com/thread/7452747 and

what Winston Churchill says about it here: https://discussions.apple.com/thread/7452747

Rcac2003 Said:

"Apple id and devices: Why does my iPhone show up on my son’s iPhone under settings under his Apple ID?"

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How So?:

The most probable reasons: Your son added it himself, or anyone else who has access to your iPhone and his Apple Account (a full assumption on my part).

What to Do:

If this was unbeknownst to both of you, then, in my opinion, it would be best if you and your son were to consider modify your own credentials:

• modifying login credentials (New eMail and New Password)
• associating a different method of payment
• enabling Two-factor authentication for Apple ID and all other accounts, that have such an option