Well you see, the reason many people describe such circuits as doublers, is because you stick in an AC signal with a peak voltage of V in, and you get out DC at 2V. This is contrary to the usual behaviour of a linear unregulated supply where it will have a no load voltage output of only V.
You can find similar circuits that can give triple or greater multiples too.
Do a web search for "voltage doubler", and you will find many circuits like that which you describe.
Now why everyone is throwing toys out of the pram over the precise naming of these circuits escapes me, since its bleeding obvious that while "voltage doubler" is indeed in very common use, there are different names and terminologies commonly in use as well.
A dubious claim IMHO, but not relevant, so let's skip that.
Yes, and calling it a doubler is also perfectly acceptable.
However don't take my word for it, see fig 1.78 and following explanation: