The full sized prayer shawl, known as a tallit, was created in order to continue the tradition of having strings on the four corners of the worn garment even when clothing no longer has four corners. The blue strand of fabric in the shawl is meant to remind the wearer of the duties of the Jewish people.
Among Orthodox Jewish communities, only men wear these shawls.
All Jewish men wear these prayer shawls in Fiddler on the Roof including Tevye who draws attention to it during the opening number "Tradition!" Although other clothes may be worn over it, it should always be worn and visible.
My Jewish Learning. (n.d.) Tallit (The Prayer Shawl) Retrieved from: https://www.myjewishlearning.com/article/tallit-the-prayer-shawl/