Quora is... ... not exactly a reliable source of information.
Is Politics.SE objectively better than Quora in any obvious, major, and uncontroversial respects, and if so, which respects would those be?
You set the hurdle quite high. Objectively better in any obvious, major and uncontroversial respect is almost surely impossible because better is mostly subjective and quite often controversial. It probably strongly depends on your perspective.
Therefore I would say that the answer to your question is a clear: No, there aren't any obvious, major and uncontroversial aspects in which Politics.SE is objectively better than Quora.
Having said that, there are of course differences between Quora and StackExchange. Both focus on Expert Q&A, but they are not the same and do it in somewhat different ways. These different approaches may have an impact on the resulting quality of the content.
I don't see what makes Politics.SE special among all the SEs, so a general reference to a Quora vs. StackExchange discussion should already give a lot of insight. Google Quora vs. StackExchange to learn more. Out of the top of my head I would say that Quora demands you to register and doesn't have the voting system of the SEs. But whether this makes Quora better or worse, I don't know really.
In addition to the license issue discussed in the comments, Quora has an annoying rule that you can't see more than one question without signing in.