I agree, it's definitely an issue with sharing Apple IDs. If you don't want to share apps, you should create a new Apple ID for one of those accounts. That account will be entirely new, so they won't have any apps. However, with Family Share, the other person can be the keeper of apps. So while two people can have two different Apple IDs and two different phones, Family Sharing lets you share apps with the primary account.
Also, I've changed your title in keeping with our quality guidelines. We had like 100 quick questions here that built over time, so trying to avoid having to go down that road again. Please use descriptive titles.
You can share iTunes ID and get the same apps and music if you want. It's in the purchased list. You don't have to download everything your son does. If anything shows, just delete it. But you need separate iCloud ID's. That's where your problem is. Also separate ID for iMessage and FaceTime.