In some states you can’t register with a party — you just register to vote and that’s it.
It’s unfortunate that in the US, there is no party that is left-leaning on economic issues but right-leaning on social or moral issues. All the religious parties in Israel are like that.
Regardless, I think Jews make a big mistake by identifying too much with a single political party, both because (given our political climate) this leads to arrogance and looking down on people who believe differently, and because it causes people to absorb anti-Torah ideologies such as those of Ayn Rand or secular humanism.