Covering the hair is the religious 'requirement' - not all follow that here. the burka isn't a religious thing at all. there's nowhere that states that a face covering is required.
It depends on interpretation of and choice of observance of religious law, whether Jewish, Christian or Muslim.
Most Christian women do not cover their hair in public, and many Jewish women also do not despite religious tenets that could be interpreted as requiring them to do so.
Until fairly recently, most Christian denominations required women to cover their hair whenever in a place of worship, and there are still some smaller Christian sub-groups that consider it a religious requirement that women cover their hair at all times when out in public. This goes back to Corinthians.
Married Jewish women, at least those who are frum, will also usually keep their hair covered when out in public, usually with a hat, beret, scarf, or snood. This comes from observing the Talmudic principle of tzniut, or modesty. Also in common use though this one is prohibited in some streams of Judaism, and this strikes me as the height of ridiculous vanity, is the use of wigs made out of human hair that some Jewish women use to cover their own hair and thus be technically in compliance with Jewish law.
These are all variations of how various Jewish and Christian denominations and the more fundamentalist sub-denominations interpret and implement their respective religious requirements of modesty and humility. They are hardly monolithic in how they do so.
Same goes for Islam. There are passages in the Quran that call for woman to dress modestly. The definition of "modest" varies between different Islamic groups and communities, but that does not mean that the more extreme forms of modesty (ie, burqa) are any less religiously legitimate than the more modest forms (light hair-scarf).