Regardless of agreement, shipping and handling are not synonyms.
In practical terms in north america, I've never seen them separate, and I buy online A LOT. So if they're not synonyms (and I agree with that pedantry:), they are functionally equivalent for most people's most practical purposes. They're just a phrase that goes together.
Ultimately, if a retailer advertises "FREE SHIPPING", and I find that I need to pay some additional fee beyond item's advertised price, to get the item to arrive to me, I will feel I've been deceived, whether they call that charge shipping, or handling, or gnorf.