I had doubted this in the past. My main reason was that Saint Mary Magdalene is called “the Magdalene” not “Mary of Bethany.” Magdala was in Galilee, not Judea, where Bethany was. I stand corrected by this informative article by Father Ryan Erlenbush on the New Advent website. I do not agree with every point Father makes, but the principal argument he develops so well has convinced me thoroughly that I had been wrong. No, it is not de fide, but it is the stronger tradition (and liturgically set in the traditional Roman Rite) that these two Marys are the same person. The saint’s feast day is coming up July 22 and her sister, Saint Martha’s, is July 29.
Many years ago, when I was in France, a priest friend took me to a church in Tarascon. In the crypt was the tomb of Saint Martha. I knew nothing at that time about the tradition of Martha, Mary, and Lazarus being put on a boat by the Jews on the shore of Palestine and that this boat miraculously landed in France. This holy family, so dear to Our Lord, also had with them the body of the mother of the Mother of God, Saint Anne.
New Advent: Although doubted by most modern biblical “scholars” and somewhat obscured by the Novus Ordo Liturgy, there is no reason to doubt that St. Mary Magdalene is St. Mary of Bethany, the sister of Sts. Martha and Lazarus. Furthermore, she is the penitent woman described in Luke 7 who wept at the Lord’s feet and drying then with her hair anointed them with the rich perfume. Full article is here.