It is absolutely possible to have a die variety that ALSO has
Machine Doubling. It's also possible for a die variety to be struck off-center or on a clipped planchet and be a die variety AND error at the same time. This goes to PROVE that die varieties and errors are DIFFERENT.
I believe there's a published photograph of a broadstruck 1972
DDO that's the size of a quarter. Although the
DDO is a die#3 and is less valuable and more common than the stronger doubled dies for the year, it's still VERY unusual for something like that to come out on an error coin.
My only "die variety/error" is an off-center example of 1951D RPM#1. It's about 20% off center, but the RPM shows nicely.
And yes, I have many examples of die varieties that are machine doubled. I usually try to skip over them or leave them behind, because to me,
Machine Doubling is coin damage and detracts from the die variety...they are undesirable.