They are.
The 'raw material' Dewey shows is a forging.
Barrels are 'machined forgings'.
https://www.youtube.com/watch?v=2Buoivte49cIf steel is poured into a mold and allowed to cool, it's a casting. If it's worked in a hot plastic state and then cooled it's a forging.
Sorry, didn't mean to overlook the obvious. What I meant to say was, "I just assumed all shotgun barrels were
hammer forged in some way."
You seem to be confused. ALL forging involves a "hammer" (of some sort) which is used to work the hot, soft metal into a given shape. Not all forging involves a hand-held hammer (unless we're talking about farriers). I know that Ithaca's cute little stamp shows a hammer and anvil but I guarantee that neither of those tools was used in the making of their barrels.
Here are a couple of photos of hammer forges.

