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 variant rules for initiative besides the above ruling: Initiative Score (i.e. a passive Dexterity check, rather than an active roll), and Side Initiative (each group rolls a d20, without modifiers), and Speed Factor (more uncertainty, less speed). For large groups of enemies, some DMs decide to split them into subgroups (see Greenstone Walker's answer to "How does having monsters of the same type go at the same time affect balance?"). 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 individuals 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).
Best Answer
I usually do this, but it can become difficult when the enemies are numerous.
This is what I usually do for initiative, and most of the time, it causes no issues at all. It is when the number of NPC combatants gets to be large that individual initiative starts to get cumbersome.
I once ran a battle against four NPCs with 8 bears and rolled them all individually, and I lost track of which bear was which after the first round. The utility of rolling the group at the same time is that you don't have to distinguish between them in the initiative order. It doesn't matter which bear is which, they all go at the same time.
This is not a game balance concern.
If anything, this eliminates a possible game balance concern. Rolling the entire group at once gives the DM the opportunity to dogpile a character without them or their allies having a turn in between attacks from the NPCs. To be clear, I never would have dogpiled one character with 8 bears, but rolling initiative individually makes that less feasible of a tactic for the DM to pull.