According to my research, the word score meant mark, scratch, cut. I have heard and used the word used in reference to marking a piece of wood for cutting. To keep score would mean to put a line or a mark for each goal made in any game.
At some point in time and in some place the word score meant or stood for a count of 20.
Consider also: a dozen eggs, tally marks. Those are similar to the word score as used with a number.
Hope this helps