Most of these similarities aren't really similarities at all, beyond the simple 'not Christian'. Furthermore, given that Islam, like Christianity but unlike Judaism is a proselytising religion, there's going to be a natural rivalry.  

Support for Israel has little to do with religion per se. US evangelicals justify their support through religion, but they do so with all their views. Instead it has to do with Western vs. non Western - Israel's dominant culture is that of the Ashkenazim - i.e. of a European cultural group, There's the Cold War legacy, the guilt over the Holocaust and more broadly the Western tradition of anti-semitism, made more relevant by the taking up of Western antisemitism by many Arabs, and the general tendency to support states over non-states except when the state in question is strongly antagonistic to Western countries (the only exception to that in recent times that I can think of is East Timor).