There's some context missing here: we are told they were disapproved of by her parents for someone as young and bright minded as her. This is not a case of a widespread social attitude, this is a case of dad not letting you read Twilight again like, ohmigod, geez. So lame. Also, keep in mind, Anna doesn't say anything about this even when describing her own education. She stresses her parents were supportive of her education. The anecdote comes to us from the Metropolitan of Ephesus, Georgios Tornikes.

Georgios gave Anna's funeral oration and was not around during her childhood. So the story is somewhat doubtful. But Georgios probably means it both as a compliment to her and her parents: her parents were so pious they refused to let her read something with themes of polytheism and immoral characters. Anna was so dedicated to learning she quietly read it anyway (and keep in mind, she would have had to study more archaic forms of Greek to understand it).

The Odyssey was well loved and preserved by the Byzantine Empire. While a consciously Christian nation, they also largely preserved things from the Classical world. For example, the Tax Free Farm was an old Athenian tourist attraction that we know the Byzantines kept alive. They largely preserved pagan temples, simply converting them to churches. This didn't extend to pro-pagan political texts, and harsh anti-pagan laws came about from the 4th to 6th century. In particular, after the fall of Rome we hear that Christians threatened to chop the hands off of anyone copying pro-pagan political texts. These laws largely remained on the books long after the pagans had disappearaed.

But the Odyssey is not a religious or political work really. The censorship laws were concerned with things like Against the Galileans, a pagan tract explicitly arguing against Christianity. Works like the Odyssey or Aeneid were not generally seen as threatening to Christian dominance.