My son and I got new phones, His phone is showing up under “devices” in Find my iPhone but not “People”

You and he must have different Apple IDs, and be signed in to your unique Apple IDs on each phone. Each of your names should appear at the top of the page when you open Settings. This is the name that will be used for People. Then, each of you go to Settings/[your name]/Find My, and verify that My Location says This Device.

No, he should set his name in Find My as being associated with his iPhone. The reason the setting is there at all is because an individual can have more than one device, such as an iPhone and iPad; that setting lets someone choose the device that they was to be associated with.