Multiclassing would likely become a default
To reward / incentivize AGAINST multiclassing, the ASI feature is based on class progression. The basic way you have more options on when to multiclass, instead of just which class to choose. It also makes it tougher to add multiple classes that require different minimum ability scores. Were that limitation lifted, the character would be able to add new classes and use an ASI to qualify for another class, allowing for more "front loading features" to accumulate. Most classes have multiple features in their first three levels.
If I allowed my players to get an ASI or feat every 4 character levels, how would the game balance be concretely impacted?
If they are solo classed it wouldn't be a change at all. Unless they get really crazy into multiclassing it won't make much of a difference. But it will absolutely lead to stronger PC's than the system intends.
How could the players abuse this?
They could conceivably create class combinations that are limited by the minimum ability scores under the normal rules which, while likely a lot of whacky fun, could lead to very strong combos. The system is that way to limit some of this because you likely have a deficient ability score somewhere to stop you from taking a bunch of classes/dips.