But all firmware is software.
Petronski Logic has left you in quite a quandary
Let's listen to Petronski's mind right now: "Daisy, daisy give...me...yo..ur....answ..ser....do"
Yet the packaging of firmware (without which it wouldn't be firmware) is considered hardware. That's why they call it "firmware" instead of "hardware" or "software" -- they needed a new term since it falls under both sets. But, unless someone is being purposely obstreperous as you are now, people will accept calling firmware either hardware or software depending on the context.