A unit of measurement is a definite magnitude of a physical quantity.
Is a year a definite magnitude of a physical quantity? What is the physical quantity? Is a day a definite magnitude of a physical quantity? What happens when we add leap seconds?
That was a quote from Wikipedia. Obviously for some things the world "physical" has to be taken metaphorically. Also, I'm not sure "physical" means "material". Time is a physical concept so I would not be surprised if it could be considered as a physical quantity, thus making the second a physical unit (few people would doubt that, anyway).
(*) Is it "number of bitcoins" or "amount of bitcoin"? "Amount of bitcoins" is grammatically incorrect. Either bitcoins are countable or bitcoin is uncountable. You seem to be treating it as the latter, in which case it would be "a bitcoin of bitcoin" or "a satoshi of bitcoin", and "bitcoin" and "satoshi" would indeed be units.
As I wrote, things would have been clearer if Satoshi had used a different word to call the "substance" and the "unit" (as I imagined, with "bitgold" and "bitgram" for instance). He did not, so "bitcoin" has several meanings. It's a bit as if we were using the word "water" both for the liquid substance and for a standard amount of it.