The text calls her Mary and nothing more. The identification with Mary Magdalene is an inference, and a reasonable one: in a range of second and third-century texts, including the Dialogue of the Saviour, the Gospel of Philip and Pistis Sophia, a Mary who questions Jesus, receives teaching and is resented by Peter is the Magdalene, and this Mary occupies exactly that role. Karen King and most specialists take the identification as secure enough to use. A minority have proposed Mary the mother of Jesus. There is no decisive evidence either way in the surviving pages, and the honest formulation is that the figure is almost certainly meant as Magdalene but is not named as her.
On its classification, caution is again warranted. The Berlin Codex is otherwise a collection of demonstrably Gnostic works, and the ascent of the soul past hostile powers is a motif at home in second-century Gnostic systems. But the distinctive machinery, a demiurge, the fall of Sophia, a creation account that makes the material world an error, is absent from what survives. Esther de Boer and others argue the text is better read against a broad early Christian and Stoic background than as an artefact of a Gnostic school. Dating rests on this and on its apparent familiarity with John and with Paul: King argues for early in the second century, Christopher Tuckett for the first half of the second century but later than King.
Its modern career has been mostly outside scholarship. It is regularly cited as evidence that women exercised teaching authority in some early Christian communities and were opposed for it, with Peter standing for the party that objected. That reading has real support in the text, since the objection raised is explicitly about a woman and is explicitly overruled. What it cannot bear is the weight sometimes put on it: a fragmentary second-century text in which Peter functions as a literary foil is not a record of an institutional dispute in the first-century church, and the missing pages mean the vision at the centre of the argument cannot be assessed at all.