Why wouldn't you allow multiclasses between a Druid and Barbarian? They don't seem that dissimilar that you couldn't come up with why you are both.
It's not about whether or not you could come up with a "why."
I suppose the simple answer is "World/Setting Internal Consistency." I play in my own homebrew world.
In Orea, the druids are a single, ancient, continent-wide and incredibly secretive/mysterious organization dating back to the dawn of history. Being a druid is a serious and consuming pursuit. Mastery of the druidic abilities, spells and powers simply does not allow room for training in/practicing anything else.
Barbarians are a human culture and a class. PCs of that human ethnic group may choose to be the Barbarian class. Not all human warriors of those tribes are actually [classed]
Barbarians and humans who are not from those tribes can not become Barbarians [class].
The idea that barbarians and druids have a lot in common/would share some ideals/get along. Sure. The barbarian culture does not have nor permit arcane magic use and the alleged "gods" of other humans are surely strange spirits or, maybe, demons! The barbarian tribes have "shamans" (I do have a homebrew npc shaman class for less civilized cultures, but imagine them as "druid-lite" for spellcasting purposes). The tribes are built around totemic animals for each tribe and it is these totem spirits that grant the shamans their magic...the shamans' connection to the totem spirits are what permit a superstitious tolerance or fearful respect for the strange magic-using men/women. So, druids are close enough/good enough guys since they're kinda like the shamans. Others with magical abilities are not so tolerated.
SO, sure druids and barbarians get along. That does not, somehow, translate to allowing cross-training.
I also prefer to adhere to a 1e-esque mode of multiclassing. The concept 3e introduced as "multi-classing" has no place in my games. There is no "level dipping" in my games. There is no PC with a string of 4/5/6 classes in my games. No ridiculous class combo's tied together with the thinnest thread of justification simply to have one's cake and eat it too. You can make the characters you want, within the framework of abundant options in the world, with all of the amazing abilities and limitations of a particular race/class. If one can not come up with something they'd want to play with the
dozens of available options of class/race combinations (including a few homebrews) in the world, then it is probably best for them to move along to a different table.
But the POINT is, in 1e-esque multiclassing, there are certain classes which simply
do not multiclass. Druid is one of those. Barbarian is another. Monks (though I have never had a player use one in a game I've run in my world) would be another. (Paladins, Bards, ...Warlocks, if one wanted to be played in the world...etc...). Personally, I would prefer to have it be, across the board, the big four only...so Fighter/Mage, Fighter/Cleric, Cleric/Mage are possible while Fighter/Druid, Druid/Barbarian, Barbarian/Mage are not...but there are exceptions, like Ranger/Cleric or Thief/Illusionist, so I can't. Best I can say, across the board, is "at least
one of the classes must be one of the cardinal/broad/big 4 classes."
So, yeah. Like I said, wouldn't allow it in my world. The option for a "Druid/Monk" or a "Druid/Barbarian" is simply not there...avoiding/negating the questions.