Israel is a Jewish state, and this is primarily in a national sense -- similar to the way that Arab states are Arab, Chinese states are Chinese, etc. Many atheists have a Jewish identity and/or support the existence of a Jewish state.

Not all Jews are religious, in fact even in Israel, the majority of Jews are secular. And Israel does have separation of state and religion, already. Calling it a Jewish state has nothing to do with religion. Seems to me like you are confused, even though you don't think you are.