Certainly a cousin could have a family resemblance...but this James was not called an "Anepsios" (cousin) in scripture. He was always referred to as an "Adelphos" (brother).
Our Lord did have a cousin named James....the brother of John and also called the sons of Salome and of Zebedee (Thunder). This was the John who took the responsibility for the care of Mary at the crucifixion. Salome is identified here by Mark.
[Mark 15:40] There were also women looking on afar off: among whom was Mary Magdalene, and Mary the mother of James the less and of Joses, and Salome
John identifies her as the sister of Mary [John 19:25] Now there stood by the cross of Jesus his mother, and his mother's sister, Mary the wife of Cleophas, and Mary Magdalene.
From this we know that Salome is Mary's sister and of course, the wife of Zebedee [Matthew 27:56] Among which was Mary Magdalene, and Mary the mother of James and Joses, and the mother of Zebedees children.
And of course we know that Zebedee is the father of James and John [Matthew 4:21] And going on from thence, he saw other two brethren, James the son of Zebedee, and John his brother, in a ship with Zebedee their father, mending their nets; and he called them.
So....yes, Jesus had a first cousin named James. He also had a brother named James....but because of his tentative relationship with his brethren [John 7:14-5] he entrusted the care of his mother to his first cousin, John, brother of this James, sons of Salome and Zebedee.
Luther, Calvin, and Other Early Protestants on the Perpetual Virginity of Mary
Luther, Calvin, and Other Early Protestants on the Perpetual Virginity of Mary