To start with, I am going to assume that the child is the biological child of both of you and that you live together. In that case, only one person can claim all the tax benefits -- dependent exemption, child tax credit, head of household filing status (sometimes), EIC (sometimes), and dependent care credit or tax-free FSA.
The Statute IRC 501(c)(4) provides for exemption of: Civic leagues or organizations not organized for profit but operated. exclusively for the promotion of social welfare. Local associations of employees, the membership of which is limited to the. employees of a designated person or persons in a particular municipality.
Non-profit organizations are run differently from for-profit organizations in that any funds they make beyond their internal expenses go to a goal they have committed to. For example, a health research organization would raise funds, pay its expenses and salaries and apply surplus funds to its cause. A non-profit's business manager oversees the structure of the organization with the cause foremost in his mind.