5e itself is bad for this style of play. Multiclassing based on story is too difficult since it's likely you don't coincidentally have the stats for it, and you can't add in new subclasses like you could prestige classes in 3e that fit the story. The lack of feat diversity is also a detriment to this style of play.
I have multiclassed for story reasons before, but you are correct it is stat-limited. It is generall a lot easier if you roll abilities though. That said I was talking about feats
It is very easy to tie 5E feats into story elements, things like Fey Touched, Shadow Touched, Eldritch Adept, the Dragon feats all have specific story elements already baked-in and many others like Magic Initiate, Chef, Skulker, Observant, Actor etc all have some implied story elements. And the thing is you can take them all whenever the story dictates it.
I adored 3e for this style of play and it's still my favorite edition because of it. 5e I really like, but is a major let down in this area.
IME is a lot easier to do this in 5E than 3E. You mention how hard it is to multiclass in 5E - if you multiclass in 3E at a high level you lose xps. If you take 2 1 level dips now you are down 40% and the prestige classes are not an answer as they all have prerequisites, again meaning you have to start "bulding the story" long before the story happens.
For example: Your good Elf Fighter is 7th level and after going through the Shadowfell, wants to incorporate that into his character and turn him into a sneaky magic assassin type. You can't take the Assassin prestige class because you aren't evil. There is the Shadowdancer prestige class, which fits perfectly, but you don't have any of the feat or skill prerequisites and you don't have a single point in hide or move silently ..... you could multiclass into Rogue to start building them, but then there is a 20% XP penalty and without bounded accuracy it is going to be 4 levels or so before you are any good at it, completely derailing your character and THEN after you get those prereqs you can start building into the shadowdancer prestige class.
Meanwhile the 7th level 5E Fighter just takes Ranger at the very next level, picks up stealth expertise and overnight goes from being below average in stealth to being a master at stealth. One level and he is already over halfway there! At Ranger 3 he has the Gloomstalker subclass, at Ranger 4 the skulker feat and then back to Fighter 8 and the Shadow Touched feat. He immediately changes his character at the next level and grows into the new vision over a few levels.
Now there are ability limits, and I need at least a 13 Dex and Wisdom to work this. But there are other options that do largely the same thing - If I don't have the Wisdom but do have the dex I can do the same with Rogue-Assasin. If I don't have the dex but I have a 13 Charisma I can even go Whispers Bard and on a poor Dex and with expertise I can still be really good sneaking around. All 3 of these can give the same sort of creepy-night flavor - sneaking around and being able to "sneak attack".
