No, children don't need their own phone.
However, in order for the child to be able to see the information in their own app, for example about their weekly or monthly allowance, the child needs to have access to a device where the app can be installed. This can be for instance be a tablet, or the parent's phone. The latter requires the parent to log out of his or her user account before the child logging in to theirs.
Please note that the child's user account also needs to be registered with an email or a phone number to be able to log in to the Gimi app.