Permitir que um bárbaro baseado em Dex faça multiclasse com base em Dex, em vez de Str, causará problemas futuros de equilíbrio?

The multiclass prerequisites, in my opinion, are best treated as suggestions rather than requirements. They do not serve any balance function—they serve primarily to help prevent players from falling into the trap of taking a class that their ability scores will not support. The reciprocal nature of the requirements, in my opinion, have more to do with making things symmetric and minimizing the degree to which the order you take classes matters than it does to do with anything concerned with balance.

As such, I have never stopped a player who knew what they were doing from doing any multiclassing they liked. This has caused zero problems, and I see no way that it could. The worst case scenario, as far as I can tell, is Paladin/Warlock using the Hexblade patron, which can now be done without needing Strength 13, which is quite desirable for this combination since Strength does little for them thanks to Hex Warrior. However, Hexblade is widely considered to be an anomaly in its own right, which means it would be quite sad indeed to limit everyone else for the sake of this one case. For me, even this combination has not caused any real problems. There is certainly nothing about a Dexterity-based Barbarian/Fighter that is going to cause more problems.

There are edge cases where not enforcing multiclass requirements can lead to trouble

A lot of this depends on using either the standard array or point-buy system for determining ability scores. If using just die rolls, then it's possible to have a character with all 18s and multiclassing requirements are moot.

The first example that comes to mind is a Barbarian that wants a level of Hexblade. I know this because I recently tried theory-crafting one. Why make this combo? Because a 1st-level Hexblade can use a Hexblade's Curse with a really great power:

Any attack roll you make against the cursed target is a critical hit on a roll of 19 or 20 on the d20.

That same power doesn't occur until third level of a Fighter taking the Champion sub-class. So the Barbarian qualifies to multiclass into a Fighter as both require a Strength of 13, but it would take three levels to get that feature. But if we ignore multiclass requirements, then it only takes one. It also means that the Barbarian is not spending points to get the Charisma to 13 and can focus on Str, Dex, and Con. But to get that 13 in Charisma, they would have to sacrifice a large amount of points from their core stats.

There are likely other situations similar to this where ignoring requirements can cause some unforeseen over powering. So while I doubt it's game-breaking, it can become a slippery slope.