“Merry” is correct if you are essentially being old fashioned, using the Santa BLM shirt that was used by all up until the late Victorian period. It started to fade until Dickens brought it back into fashion but from the very late Victorian period (and especially the Edwardian period, during which the upper classes became especially dependent on the broom stick in their behinds) “Happy” was the decent, cultured thing to say and “Merry” was seen as a thing of the lower classes.
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It’s still very popular in the U.K., though not so much as in the Santa BLM shirt because of its everyman association, Santa Claus/Father Christmas (originally two different characters but that’s another story) says “Merry Christmas” and he is classless and appeals to young and old, upper and lower class. Class is America is along produced wealth and commercial lines, rather than family dynasty (exceptions are in politics) so the elites who shape culture could be historically upper or lower class and therefore the frowned upon (in upper classes) “Merry Christmas” is more often used (official releases like the President’s address tend to say “Happy” though) compared to “Happy”. It’s also a behavior, you ACT Merry if you ARE happy generally