I believe that Shams was indeed Rumi's Twin Soul, and in that profound awakening, he discovered the union we all have with the One, whom he called the Friend. Once that happened, Rumi's soul opened up and he recognized that we are all part of the One as drops of water are all a part of the sea.
I think that Salah and Husam, Rumi's later spiritual companions after Shams' death, embodied that recognition and he celebrated that in his later poetry.
Perhaps that is the purpose of any true Twin Soul relationship: to awaken us to our soul's connection not only with our other half, but to the divine and indeed to all in the All. Enjoy this:
"If you are me and I am you,
What is this separation between you and me?
We are the light of God, we are God's mirror.
So why do we struggle with ourselves and with one another?"