actually, score starts out meaning scratch or mark, moves on to tally (scoring a game,) and then from there to meaning a count of twenty and by another path, a musical score (a tally of all the notes as it were!)

the meaning of score has meandered.. but one word, one root.. all you have to do is check out a dictionary.

even a simple M-W 10th has two entries for case, with some brief history, and one entry for score.. and progressing meanings..

this is really a simple game.. and you can check things out for your self..