You misunderstand the "separation of church and state". That phrase means that the church will not make laws, it does not mean that the government will pretend that churches do not exist. In America right now, it is beneficial for a politician to be a Christian (or at least pretend to be one) so they use a Bible.

If they are another religion then they can use a different book. Keith Ellison is the first Muslim elected to Congress and he was sworn in using Thomas Jefferson's copy of the Qu'ran.