No, not dressing
as a Muslim, but dressing
like Muslim women do today.
But I really love your turn around…“Why do modern day Muslims dress like the Mother of God?”

ha, ha!
Many modern Muslim women wear scarves or other types of head covering. While I am not 100% sure of their reasons, I imagine it has something to do with modesty.
Married women that are Orthodox Jews, to this day, also use some sort of head covering. While scarves are sometimes used for this purpose, many of them also use wigs. A Jewish friend of mine has explained the reasons for this practice ~ when a woman is married, according to Orthodox Jewish tradition, the only person that should ever see her hair uncovered is her husband. This tradition is related to the attractiveness of a woman’s hair. This same tradition would have been practiced in the time of Mary, so her practice of covering her head or her hair is rooted in Orthodox Judaism. Since she was a married woman, she would appear with her head covered.
When I was much younger (I’m a Grandma now) it was traditional for a woman to wear something on her head in church, especially in a Catholic church. I didn’t grow up Catholic, I was Episcopalian then, but EVEN THERE I remember as a teenager that no one would dare go into either our church or a Catholic church ~ whether it was St. Peter’s in Rome or the local Catholic parish church ~ without some sort of head covering. Short lace mantillas and head scarves were a common choice for this. Hats were, of course, OK too.
Then I guess we got “women’s lib” and everything changed. But some women in my parish still cover their head. And if I were lucky enough to get to go to St. Peter’s in Rome, I would be happy to do so too. Which I supposes raises the question why I don’t do so now. But that’s another discussion.
As I said before, I cannot speak for the thoughts of Mary, but since she spent her life setting an example for the rest of us to follow, I suspect that her modest dress, and perhaps even her practice of covering her head, is a message to the rest of us. But even if it wasn’t meant to “send a message”, this is the way that apparently she prefers to present herself. Perhaps she does this to be recognized for who she is. If she showed up in shorts & a tank top, maybe people would wonder.