Because mass is measured in grams, milligrams, kilograms, etc. and grams is the sensible choice considering kilograms would be a ridiculously tiny number.
Also, grams is the "base" measurement, like a unit of 1. Milligrams would look more like a normal number, but grams compares it to something that weighs, well, one gram, which is a lot more known to us. (ie we can lift an object and say it's about a gram, but it's a lot harder to tell if something is one milligram because you would barely even realize you're lifting something.)
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Mass
Yes
You use grams to find the mass of an object.
Grams are the measure in this case. A gram is a unit of mass.
No. Cubic centimeters are units of volume. Mass is measured in grams, and the standard mass is a kilogram of metal kept under the strictest conditions and tightest security in France.