For fear of stating the obvious, but Santa Claus is based on St Nicolas of Myra, who was the Bishop of Myra (in what is today Turkey). Various miracles are associated with him, but maybe the best known is that during a time of famne, an evil butcher killed three children and hid them in a vat, intending to sell them as meat, but that St Nicholas found their bodies and brought them back to life. Thus he became the patron saint and protector of children. Quite possibly the story is apocryphal and based on the misunderstanding of an old illustration. The illustration actually shows the saint baptising three adults, and through the artist’s incometence, the baptismal font looks like a meat vat. The real Saint Nicholas’ life mostly consisted of teaching and spreading the faith and he baptized very many people.
It is of course entirely correct to depict him in bishop’s attire because he was a bishop.
In many countries Saint Nicholas is still depicted in bishop’s attire today. For example in Holland and in many parts of Germany and Austria. In some parts he is accompanied by an assistant or evil countrpart. In Holland this is the Piet, who is depicted as a Spanish page, often with dark skin, which is leading to a lot of controversy of late. Further souith he mutates into the far more evil Karmpus, or Grampus, a hormed amd hooved fire demon from hell. Typically one or several people wear krampus costumes and parade through town. The coming of Saint Nicolas is the driving away of the evil Krampus.
The symbolism is connected to advent, and of light driving back the darkness in preparation of the birth of Christ. The person of Saint Nicholas has thus become attached to a different tradition, which has little to do with the Nicholas of history. But it’s not entirely misplaced as a bishop’s mitre is of course modelled on a shepherd’s staff, used both to prevent sheep wandering too far bit also to ward off wolves and other wild animals who want to harm the sheep. Nicholas thuis protects the faithful, and especially the children, against this evil wolflike figure.
This has nothing to do with Christmas by the way, as Saint Nicholas day is on 6th December, and still celbrated on that day in these parts. I don’t know how that somehow got rolled into Christmas.