What is an appropriate name for a property that tells whether a list is empty, single or multiple?
The nearest term you'll probably find is the hypernym cardinality.
cardinality [noun] [mathematics]
The number of elements in a set or other grouping, as a property of that grouping.
[Lexico]
This takes values 0, 1, 2, 3 ... rather than 0, 1, many.