Analog, meaning non-quantized (i.e. capable of smooth variation), has become the word most often used to express a contrast to digital by default, even in cases where it may not be strictly appropriate. (It often forms the modifier in a retronym, a new word or phrase used to distinguish the original version of something from a newer rival.)
Analog probably became the word of choice after its use in analog recording (as opposed to digital recording technologies such as CDs) and analog watch (as opposed to digital watches). In both cases it was appropriate because the smoothly varying analog signal was a direct contrast to the quantized digital experience.
In this particular case, even non-digital money is quantized, but analog at least has the advantage of being readily understood.