pair, set, comparison, counter for pairs
雙 centers on the idea of two matching items forming a unit: a pair, a set, or a couple. The comparison and counter senses grow naturally from comparing or counting such pairs.
雙 is the traditional form of 双. It combines two 隹 (short-tailed birds) above 又 (hand), originally depicting a hand holding two birds, representing a matched pair.
Two identical birds 隹隹 are held in one hand 又, making a perfect pair. Picture a hunter proudly showing off a matched set of two birds.
For ソウ, imagine sewing a pair of matching socks: sew -> ソウ, and each stitch creates a perfect pair.