[invalidname]

Israel doesn't have a constitution. There's the declaration of independence which does explicitly say the Jewish People. But the interpretation of that is varied. The closest thing Israel has to a founding father (who died before the country was formed) is Hertzel and he called it "state of the Jews". Many make a strong distinction between that and "Jewish state" as American TV anchors like to say (that's like nails on a blackboard for me).

Israel doesn't have civil unions. It only has religious marriage any religion is OK but the religions do discriminate. This is both problematic and a blessing. I'm not married to my spouse because I'm an atheist, but Israel has a "known in public" status which gives us the same rights (and some obligations) as married people. That means Gay and inter-religion couples can enjoy all the rights.

Another option some people take is to fly to nearby cypress to have a civil union which is recognized in Israeli courts.

About the broader question. Yes, there are religious elements in the government that have been pushing the secular envelope. This creates a lot of friction within Israeli society. E.g. there's a law against pastries in passover which is just the dumbest thing ever... Unenforceable and just stupid. There's also regulations prohibiting Jews from working on a Sabath (supposedly as a labor protection). Notice that both laws mostly affect people who are listed as "Jews" even if they are secular.