Q: My son's phone calls him by my name when using Siri. Anytime I change it for him, it changes it on my phone.

If you share an AppleID for iCloud, iMessage and FaceTime then you are sharing one account each for each of those services. Contacts are synchronized via iCloud. Family sharing is only about sharing purchased content from the App and iTunes stores and has nothing to do with iCloud, iMessage or facetime.

You and your son each need your own AppleID to use with your own iCloud, iMessage and FaceTime accounts so you don't get collisions over sync'd settings and content.  If you need emails to create an AppleId just get free google, yahoo, AOL or whatever free email accounts you like.

You can the use family sharing (using those AppleIDs) for purchased content while keeping everything else separate.

The are some special circumstances, limitations, and restrictions you can apply to your child (under 13) with Family Sharing:

Family Sharing and Apple ID for your child - Apple Support

As the above article states:

Manage your child's account

Children enjoy the same Family Sharing features as other family members and, by default, can access the same music, movies, TV shows, books, and apps. You can limit what content your child can access on the devices they use by setting Restrictions on an iOS device or Parental Controls on OS X and iTunes. Ask to Buy is enabled by default for children under 13.* If you don't want to use Ask to Buy, you can turn it off.

Your child must remain part of your family group until they turn 13.

You might also be interested in:

Set up Family Sharing - Apple Support

AS far as family sharing for purchased material, see Family Sharing and Apple ID for your child - Apple Support

On his iphone you can use restrictions and keep the restrictions password to yourself - Use Restrictions on your iPhone, iPad, and iPod touch - Apple Support so you can disable features or limit functions as available in restrictions.

You can do this without the factory reset.

Just get him his own child's Apple ID and sign out of all services on his phone first. Once you sign back in with his Apple ID, you will be independent of each other.