... except that in the specific case of marriage equality, she was approximately always for civil unions that were fully legally equivalent to marriage, opposed in 2004 a proposed Constitutional amendment to define marriage as between a man and a woman (even though the amendment would have permitted civil unions), and "changed her mind" (publicly endorsed same-sex marriages) in 2013, well before the campaign.
In particular, she's a United Methodist (as am I; I go to an ELCA congregation these days but the two denominations are in full communion), and the United Methodist Church still teaches that homosexuality is contrary to Christianity and does not officiate same-sex marriages. (Yes, the ELCA does do same-sex marriages, and no, I cannot explain how this interacts with full communion.) Going from "my religion teaches that it's wrong" to "... but that's just my religion, and the government should give same-sex couples the same rights, but the word 'marriage' is about religion and tradition" to "I support it" is a journey that lots of non-right-wing religious folks in the US, myself included, have gone through in approximately the same time frame as when Hillary did. I think Occam's razor should push us in the direction of believing that she's genuine, or we would otherwise need a reason why she's fake but lots of people of extremely similar religious and political positions have not been fake about the exact same thing.
Going a bit more broadly, I would argue that it is common for people to change their minds on political issues in lockstep with the median of people of similar political and cultural leanings, sort of by definition.