Yeah, but applied to counters it would be the symmetric difference between multisets, which doesn't have a natural definition. If I understood the proposal they'd be defining it as absolute value of the difference of the counts, which isn't even associative.

If they only considered parities it could be interpreted as addition in F_2, which is more natural, but I'd still agree that it's hard to see how you'd use something like this in practice.

You can get the L_k distances between the two counters. E.g. if you sum the absolute value of the difference of the counts, you get the L_1 distance between the counters. If you raise them to the n^th power and then sum them, you get the L_n distance. For n=2, that's the Euclidean distance (squared).