I've always disliked multi classing for a variety of reasons but I've never forbidden the practice in any game I run. I came to dislike multi classing in 3rd edition because it was too easy for players to come up with ridiculously broken combinations. And you had a few classes that people just dipped their toes in for some of the benefits that came with it at level 1.
But as to the OP, most of my players tell the other PCs their class during introductions. I don't really see it necessarily as in game though. Oddly enough, someone can be a sorcerer, wizard, or warlock in game but nobody's really a fighter. You might be a knight, a warrior, man-at-arms, gladiator, etc., etc. but almost never a Fighter.