In other words, the idea that, from a Shinto or Buddhism perspective, no distinction is made between whether you visit a shrine (a Shinto place of worship) or temple (a Buddhist place of worship) remains today, and this applies not only to hatsumode, but to other cases as well.
つまり、初詣に限らず寺社への参詣に神道･仏教の区別は無いとされていたことの名残である。