The rules for initiative in PHB read (emphasis mine):
Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time.
The DMG offers several alternative options to the above ruling: Initiative Score (i.e. a passive DEX check), Side Initiative (each group rolls a d20, without modifiers), Speed Factor (more uncertainty, less speed). For large groups of enemies, some DMs decide to split them into subgroups (see this answer). I find the ruling about initiative in PHB asymmetric$ ^1$ and none of the listed alternatives really satisfies me.
Would a house rule that allows rolling initiative for each$ ^2$ enemy (even of the same type) be a game-breaking one? Or would there be just minor problems at most?
$ ^1$ I wonder why the PCs can have all different initiative rolls while monsters (even of the same type) should share the same one. They are still different creatures which may react with different speed.
$ ^2$ Do not consider here the issue of rolling a lot of d20s for the initiative (if you consider it a problem).