If I agree to shared parenting do I have to agree to let him claim my son as a dependent every other year?

No you don't, however, which parent can claim the child depends on IRS rules. Bascially the parent that provide more than half the support gets that. Generally it's the primary custodial parent.

A lot too depends on your shared parenting plan. But mostly you need to agree on who will claim. It would be best if you trade off year for year that the parent that doesn't have the child that years get to claim them. That way you see the refund when you have the child and when its most needed.

The high time (>50%)parent is entitled to the dependency exemption as a matter of law but it can be released by court order or an agreement of the parties. If the custody arrangement is 50-50 then the exemption goes to the parent with the higher AGI (adjusted gross income) again subject to being released.

It is best if agreed to in the custody agreement, so it is not a issue fought over every year,

It is very very common to be done every other year,