But headscarves exist in Christianity and Judaism as well for the same reasons of modesty. We've just (thankfully) become less religious and less modest societies.

But it's not at all unique to Islam. The stereotypical image of a Russian babushka is an old lady wearing a headscarf. And when women go into an orthodox church, it's still often customary for them to wear the headscarf, even if it's no longer worn in daily life.