There is a textual problem here that most readers never encounter, and it is a good example of how manuscript evidence works.
A small number of Old Latin manuscripts, three, and they are early, attribute the canticle not to Mary but to Elizabeth. Irenaeus in one place appears to know a text reading Elizabeth, and Nicetas of Remesiana refers to Elizabeth as prophesying the song. The overwhelming majority of manuscripts, in Greek and in Latin, read Mary.
The argument for taking the minority reading seriously is contextual: the canticle echoes Hannah's song, and Hannah's situation, long barren, then granted a child, is Elizabeth's situation, not Mary's. A scribe who found Elizabeth in the text would have every reason to correct it to Mary; a scribe who found Mary would have no reason to change it to Elizabeth. That is the harder-reading principle, and it points the minority way.
The argument against is the sheer weight of manuscript evidence and the fact that the surrounding narrative flow reads more naturally with Mary.
The scholarly consensus favours Mary and has done for a century, and the minority position has had distinguished defenders including Adolf von Harnack. It is a live question rather than a settled one, and it is not a controversy invented to unsettle anybody, it turns entirely on a handful of Latin manuscripts and on how one weighs the difficulty of a reading against its attestation.