No, that's the myth. I'd guess that whatever forging process was used was mostly of benefit to drive out impurities. A sort of super slow, hammer powered Bessemer process. The soft & hard blade thing is a result of heating and cooling processes which aim to impart a martensitic, hard, sharpness holding edge while the body of the blade remains a non-brittle form of steel. That's what quench and temper and case hardening achieve. The folding thing, although perhaps of benefit to condition whatever pig iron the stack furnace yielded, is not in itself a process which would benefit an already chemically pure carbon steel.
You can see the difference in things like a metal file which can file away metal from a hammer head and not get dull but which would break if you hit a nail with it. The difference is just the crystalline structure of the steel that gets formed depending on heating and cooling practices. You can make a file, or you can make a hammer, or you can make a hammer-like body with a file-like edge if you have steel with an appropriate carbon content and you heat and cool right. It doesn't make a difference if you fold it or not.
