TennisPoint accounts are designed for a single player so this unfortunately means that a family will need separate TennisPoint accounts for each person. This also ensures the correct name on the match scorecards to keep the opposing captains from having any issues with a "non-registered" person playing. Normally each family member has different match availability overall as well.
You can sign up your child's TP account here: https://www.tennispoint.com/register/user
Can the family all share the same email address? Yes, all family members can use the same email address. When registering the second person with the same email, the registration page will show a red message about the email existing on an account already -- please ignore this message. Key thing to remember is that each account's username must be unique.
Any way to link my child's account on TennisPoint with my own account? Yes, you can link your child's account to your personal TennisPoint account so that you can easily bounce between accounts!