Hard to say what the symbolism of swans were. In northern legends ‘the seven swans’ were about a girl’s seven brothers being turned into swans and she had to save them by doing something, I forget what. Then there is the northern swan constellation in the Milky Way.
By knitting them shirts made from stinging nettles. She also had to keep silent until they returned to their normal forms. Because of her silence she was accused of being a witch.(Even though the prince had fallen in love with her and pledged a troth to her.) She was sentenced to be burned, and as the stake was to be put alight her brothers came to rescue her. She quickly threw the shirts over their forms. One shirt was not finished so the youngest brother had a swan wing instead of an arm for the rest of his days.
In some tales it is 12 brothers.
I do not know about this, but:
“Seven swans a-swimming” is supposed to be about the Gifts of the Holy Spirit.