The implied point is false. First, it's a different era and the norms have shifted. Second, HC put herself forward as a feminist, and rightly or wrongly people might assume that would necessitate a less tolerant, more "independent" (I will leave him) attitude. And so her not doing so could be painted as a hypocritical effort to instead stay close to political power (because, depending on the era we are talking about, she was indeed unusually heavily involved in the White House, politics and campaigns, and eventually her own political aspirations).
Whereas for MT the issue would merely be about the more typical assumptions concerning lifestyle, her child, family values and so on. I don't think it's plausible for someone to regard her decision to stay with her husband as related to the exercise of governing or political power or to set her up for that in the future.
So I think the meme is trying to leverage a specious set of assumptions.
On the other hand, to the extent it means to allege there is a hypocritical asymmetry in how the right cherry-picks what it will call out, and that it does so more on a basis of convenience that intellectual consistency, I think that despite the failure of the cases to be parallel and so the failure of this to be such an instance, the overall accusation is correct. I am not sure this was true in the 1980s and 1990s. There was a stronger intellectual/principled line of through underpinning the conservative movement. It's now a populist movement that doesn't have pretence of intellectual foundations.
Furthermore, of course the meme has sexism and other problematic assumptions in the background, insofar as these matters are seen as matters of public consumption at all, and that women (in a way that doesn't seem to be the case for men) get to have their intimate choices second-guessed as a public sport, and that woman are seen as failing or insufficient in some way if a man cheats on them, and that their reaction to it is a test of their character in the eyes of the community, and so on.
And I did a steelman because AFAIK reporters didn't actually ask HC that question much. But I assume it refers to a broader sentiment in the culture, in which people wondered about it and editorialists and analysts discussed it.