Dual family taxes

There is no way for you to confirm that and the IRS will not tell you.

The child can only be claimed on one tax return.  Duplicate use of the child's SSN would result in the IRS rejecting an e-file or investigating to see which parent was the custodial parent.

The IRS will not tell you if someone else tried to claim your dependent.

If the other parent claims the child and your e-file is rejected, then you can file by mail and let the IRS sort out who can claim the child.  

The other answers are correct. Besides, his tax return does not affect you, so let it be.

Q. I suspect my stepson's father keeps claiming him as a dependent even though he lives with me year round. How can I confirm this?

A. By filing a competing return (presumably a joint return with the child's mother), claiming the child as a dependent.

If someone else claimed your child inappropriately, and if they file first, your return will be rejected if e-filed. You would then need to file a return on paper, claiming the child as  appropriate. The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your child was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.

Winner gets the tax benefits; loser gets to pay the IRS back with penalties and interest.  The custodial parent almost always wins. The IRS goes by physical custody, not legal custody. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree.