Yeah, allowing some limited multi-classing where it makes sense for companions would be okay probably, but fully changing classes would actually break the story, especially since for some of them their class is tied directly into their stories. Shadowheart being a cleric of Shar is her big thing, as is Gale being a wizard and Wyll a Warlock. Astarion probably has a bit more wiggle room since "rogue" has always been kind of vague out of world, but even then you can only change him so far before the story breaks completely.