Never having been a smoker….
The type of paper does not really matter, within reason, if you measure it first: I use a strip of ordinary printing paper, and "glue" it to the work with oil.
For other workshop purposes, expired plastic bank or club membership cards can be effective shims (measure first – but they appear of highly-accurate thickness). For bank cards, cut off the strip and embossing first, as instructed anyway. I set the magnetic readers on my mill's DRO using a "feeler-gauge" cut from a previous year's association card: non-magnetic.
….
I think the fondness, or preference, for cigarette paper is purely historical. The writers of the earlier model-engineering reference-works were great craftsman but still had to relay practices based on what they and most of their contemporaries had in their own workshops and homes. Besides, in their day far more people smoked than now.
Maybe someone of our next generation will gaze fondly on his, or her, Gold Medal winning, IMLEC-champion 7-1/4" g. 'Britannia'; and describe how to set up the ubiquitous 'Myford' 5-axis NC machine, using a plastics-free, nicotine-substitute, vapour-oil bottle…