2- All units are equal to a constant.
So, what constant is a dollar equal to which makes it a unit?
Presuming you mean the US Dollar 371 4/16th grains (24.057 grams) of Silver when it was established. This was done by taking a bunch of the silver coins then in circulation, measuring them and averaging.
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All this aside, bitcoin is a unit of account, in the same way that all other currencies are a unit of account.
Just because it's value and purchasing power change does not make this untrue. Time, date, payment, all are recorded in the block chain so it makes for a better unit of account than any other currency commonly traded.