Merriam-Webster defines madame as:
[...] a title equivalent to Mrs. for a married woman not of English-speaking nationality [emphasis mine].
I suspect this dates back to the days when French was the official language of international diplomacy, which began around the 14th century all the way to through post-WWII period when English started taking over. You do still see remnants of this literal lingua franca in settings like the Olympics, where all announcements are made in English, French, and the language of the host country.
So Jiang Qing, the wife of Mao Zedong, was often referred to as Madame Mao, because this would have been the form of address in formal diplomatic settings. Again, French isn't the dominant diplomatic language today, so you'd probably be less likely to hear Madame used for a contemporary political figure.
Even putting aside the decline in the use of French, when talking about internal politics, the rules of international diplomacy shouldn't apply. As English-speakers, we should use Madam as an honorific for an English-speaking person.
I could imagine Madame being used as a joke to invoke the old-fashioned, very formal international diplomacy mood in a sarcastic way. If done deliberately, it has a little bit of the same effect of calling a non-royal person "your majesty" sarcastically, although it's not nearly as strong or insulting (a lot would depend on tone of voice if it were spoken). This is one reason that people are drawing attention to this particular usage by the President - it does have a possibility of a tiny hint of sarcasm, although it could also just be an unintentional error.