For sure not everybody but sure some do and for sure if you can make people think "this looks like the original stuff but at a nice price" that is a boon for your sales. You do not need to lie for this, often just having a not crappy design for the packaging is enough, since people often (and rightly so) don't really know what they want.
As an Italian we can get a bit emotional on this (and I truly think that some imitations are truly worthy of their own respect) but there is value in distinguishing a DOP brand, eg it make manufacturing much more accountable for the quality of their products.
Interestingly enough, the only Parmigiano-Reggiano on the shelf was BelGioioso. I went "oh shit, they were right, there is some tomfoolerly in advertising." Then I noticed they actually were importing it from Italy. They had a block of Parmigiano Reggiano right next to a block of Parmesan from WI.
Once I looked at the two though, I dont think anyone could make the claim they were trying to be tricky. The Parmesan was labeled Parmesan. The import was all fancy, had IMPORTED stamped all over it, they were using it as a selling point to differentiate between the two.
As for the results, the BelGioioso Parmesan was easily the worst of the three. BUT, it was also half the price of the Satori Montamore Cheddar/Parmesan and Satori Sarvecchio Parmesan.
Im not sure if every times a brand describes a cheese as Parmesan they are trying to be misleading or trick people into thinking its from Europe, they are just describing it in a way that aligns with Americans expectations.