CAIRO — There is a strong undercurrent of competition in Egypt these days, an unstated contest among people eager to prove just how religious they are. The field of battle is the street and the focus tends to be on appearance, as opposed to conviction. It is not that the two are mutually exclusive, but they are not necessarily linked. As Egyptians increasingly emphasize Islam as the cornerstone of identity, there has been a growing emphasis on public displays of piety. For women, that has rapidly translated into the nearly universal adoption of the hijab, a scarf fitted over the...