This is a matter of Federal and State Taxation Law, and not necessarily a law created or governed by State based Divorce and/or Child Support Laws. However, in a Divorce Order, or in a Child Support Order, the matter of which parent is entitled to claim the child or children on the yearly tax returns can be assigned freely between them in a Settlement Agreement or Ordered by the Judge in a contested case.
One needs to make sure that the issue of claiming the child is clearly set out in the settlement agreement or the court's final order. If that is not done, then it may be unclear to the parents, who gets to claim the child.
The IRS rule is whichever parent provides more than 50% of the cost of living for the child has the right to claim the child. Child support paid rarely rises to 50% or more of the total expense.
Typically, it is the physical custodian who receives child support that actually provides more than 50% of the funds used to support the child through the course of a year, and thus would be entitled to claim the child on their tax returns.
If it happens that both parents claim the same child on a single year's tax return, it is typically the person that filed their taxes latest in time (or second) that will be contacted by the IRS. The automated systems employed by the IRS are unlikely to miss the fact that one child has been claimed by two separate tax filers in a single year. No one wants the IRS calling on them.
The parties can agree, and the court can order that the parents alternate years claiming the child. However, in a contested case the court is more likely to award that right to one parent for all years.
Child support is not income to the recipient and is not a deduction for the parent paying the child support.
The deductions for minor children can be substantial and this is an issue that should be addressed by the parties to a Divorce Case or a Child Support case. Particularly for low income parents when the Earned Income Credit is a part of the mix.