No, it is not. A gram measures mass (like ounces or pounds), and a cubic centimeter measures volume (like cubic inches or quarts). They do not measure the same kind of value.
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No. One gram is a measure of mass only. One gram per cubic centimeter is a measure of density, which is mass/volume. One gram per cubic centimeter means that every cubic centimeter has a mass of one gram.
A gram is a measurement of mass. It is the mass of one cubic centimeter of water.
A milliliter is the same as a cubic centimeter.
A cubic centimeter is a cubic centimeter always and no matter what! It doesn't matter what the temperature is! However, I think you are asking for the mass of a cubic centimeter of water 4 °C and that is exactly 1 gram.
They have the same volume but not necessarily the same shape.