It's all about feedback and communication with the player: there has to be something letting the player know how they are performing and that should not be the results screen.
It can be implemented in many ways like some have said above:
1. Different flashing colours for bosses (white when being damaged, red when near death). It's certainly a good way of keeping the player's eyes on the boss instead of keeping watch over the boss and a health bar, though it tends to clash with more realistic artstyles.
2. Damage numbers and health bars are useful when it's not feasible (financially or technically) to display actual damage or even particular status effects.
Eg. Imagine musou trying to convey to the player which enemies are actually cut, bleeding, bruised etc.
3. Having the enemy's appearance display that you've done damage like in Monster Hunter. This works because MH enemies are almost all large enough to visibly display the results of your efforts.
It's also part of their design where you're expected to get familiar with each monster. Not having a super granular damage system like typical RPGs also means that the player can more easily internalise how many hits they need instead of eyeing pixels and numbers.
4. It doesn't have to be limited to just visuals. Having good sound effects and dynamic music (mainly for bosses and their phases) will help tell the player that things have changed.
Some games might even have a sidekick to literally tell you the enemy's status, like MHW's Handler in the early stages.
5. Also like mentioned above, sometimes the health bar/s is useful for conveying a different message. Multiple health bars makes you think "this isn't even his final form". Final boss of MGR has a bar that reads 200% instead of 100%; you're meant to think shit has gotten real.
In short, it's a case by case basis and when implemented thoughtfully can enhance (or sometimes purposely, temporarily hurt) the player's experience.