If you save to become immune to a creature's trait/ability, are you immune to that trait/ability from all such creatures?

If you fail a save from a Pit Fiend's Fear Aura, you are Frightened

Yes, you have to make a save against each individual creature that has the fear aura.

No being immune to one fear aura does not make you immune to the other fear auras.

However.. If you look at the description of Frightened, you will see that the effects do not stack. If you have Disadvantage, then you just have it. It does not stack. But you would be unwilling to approach each individual cteatute who's save you have failed.

A frightened creature has disadvantage on ability checks and attack rolls while the source of its fear is within line of sight. The creature can't willingly move closer to the source of its fear.

Note: the combined spell effects does not apply here because the aura has a particular effect specific to each creature. Which is that you can't willingly move closer to that individual.

The Pit Fiend’s fear aura ability reads:

If a creature's saving throw is successful, the creature is immune to the pit fiend's Fear Aura for the next 24 hours.

You would only be immune to the particular pit fiend’s ability. An ability explicitly states if saving against it immunizes you against the same ability from all creatures of that type. For example, the Krasis’ hypnotic display ability says:

A creature that succeeds on a saving throw against this effect becomes immune to the Hypnotic Display of all krasis for 24 hours.