Let's Play some ProSet!
Python 2, 60 bytes



p=[0]
for i in input():p+=[n^i for n in p]
print~-p.count(0)


Try it online!
This generate the powerset from the input (based on this answer), reducing each subset by xor.
The output will be the amount of 0s (0 = all bytes/dots appear an even number of times).
In fact the xoring is done instead of the "powerset appending", but the result is the same