Transitioning from a funny, engaging story about someone's cousin's roommates' sister's wedding and the greased pig that got loose at the reception to a similarly engaging point of interest about Aumann's Agreement Theorem and how it applied to a recent decision to donate to a particular charity.

But if you used the phrase "That's exactly like", I think it would sound equally forced. Or at least it would to me.

Given that the topic being considered "impolite" implies that most people won't like most possible conversations on the topic,

I don't think it does. People generally love it if you agree with them - it's impolite because you don't always agree. Many people are happy to drone on for hours about their political beliefs, which is what makes it impolite. If you can be respectful about politics, I see no reason not to bring it up (if you want to). And if the other person can't be respectful, then that gives you (in my book, what would be) a significant reason to not pursue a relationship with that person.