No, who is declared as a dependent on taxes does NOT have to match who is so claimed on the tax returns. With divorced families, it is not unusual for splits to be agreed upon for maximum tax advantages and the IRS allows this leeway. That way both parents, can say, claim HOH, by each claiming a kid as a dependent on their taxes, even if the decree and actual living arrangements and support are such that only one parent is truly custodial for both kids.
For FAFSA, its a whole other definition that goes into WHO is the custodial parent. It doesn’t matter a whit who is claiming the child for tax purposes, or what the divorce decree says. The custodial parent who is the one who files the FAFSA is the one with whom the student lived the most in the 365 days before the form is filed. If the tiem is split evenly and can’t even have one parent claim one more night the kid spent, then it goes to who provided the most support to the kid. It doesn’t matter whether you lose out on PELL or not.
So if your student spends about equal time, then make sure the that it’s at least one more day/night with Dad. There won’t be flags with discrepancies between that and the IRS form, but if Dad lives in Alaska and you live in Maryland, and your DD is going to school in MD, there would be a question as to how the heck she’s spending more time with him.