"Bar" is not a Kurzzeichen, Carullus.
Not as such, no, but your examples (mm, km, mg...) were. They are obviously written in lower case, whereas the underlying units (Millimeter, Quadratkilometer, Milligramm) are not.
The German word is "das Bar."
Quite so, and if you are talking about the unit as such ("Das Bar hat sich als Standardeinheit des Drucks etabliert") you'll capitalize it, but in a sentence like "1 bar entspricht etwa 750 mmHg" (true, by the way) you'll write it in lower case, using it as Kurzzeichen, even though it is not an abbreviation in the strict sense.