To get the total number of unique label pairs, you could get all label names via /api/v1/labels, and then for each label name, you'd have to run the query:
count(count by(<labelname>) ({<labelname>=~".+"}))
...and then sum up all those results together.
I don't think we have a handy way of getting the average bytes per label pair, but you could again get all label names like above, and then manually process the label pairs that appear in a query like this:
count by(<labelname>) ({<labelname>=~".+"})
You could write a small script / program that could do this via the API I guess.