As you astutely note, there is no "character" reason to permit it other than for "min-maxing" or, really, and more accurately, "power-gaming." The only reason players would argue FOR multi-classing is because they want to have their cake and eat it to. They want to cherry-pick their character abilities and they can't do that without an MC level of having access [eventually] to alllll of a class' features.
The assertion that there might be "story" reasons to allow it...that's kinda...well, bogus. There would only be an "in story" reason for it if the DM has such a story to take place. There are a million OTHER ways to handle some story situation/reason then "sure, go ahead and start up a whole second class."
From the player side, the argument from a player/background "story" perspective, e.g. "I've got this GREAT idea! So, she was a born Sorcerer and then joined up this knightly order and took an Oath of Vengence against the demon that killed her brother & sister-knights. She's the last of her order. So I'm a full Pal/Sor [with all of the bells and whistles, thereof]." So you have to allow MCing, because the player has come up with this story? *shrug* And, it's simple, no you don't.