It depends on the wording of the source of the effect
The key here is to separate cause (the spell/ability) and consequence (the Frightened effect).
Let's take the situation one step at a time :
You are in your paladin's aura. You gain its effect, meaning you cannot be inflicted with Frightened.
You become the target of a spell or ability. Nothing says you are immune to spells or abilities, so nothing out of the ordinary for now.
The spell/ability asks you for a saving roll (if it doesn't, go to the next step). Nothing says you are immune to saving rolls, so you roll.
If you fail the roll (or are not asked to do one), you are now under the spell's/ability's effects.
What happens next depends on the wording of the spell/ability. There are two ways the wording of the spell/ability can make it obvious :
If the wording is something like "While you are under the effects of [this spell/ability], you are Frightened." then the effect is "suspended". It will not affect you while you are in the aura of your paladin, but if you leave the aura while still under the effect of the spell, you will be inflicted with Frightened.
If the wording is something like "When the spell takes effect, you become Frightened." then the effect is only applied once. Since that application fails, you are not afflicted with Frightened, and can leave the paladin's aura freely.
Let's get back to the example of the question. The Frightful Presence ability states the following (emphasis mine) :
Frightful Presence. Each creature of the dragon's choice that is within 120 feet of the dragon and aware of it must succeed on a DC 16 Wisdom saving throw or become frightened for 1 minute. A creature can repeat the saving throw at the end of each of its turns, ending the effect on itselfon a success.
Here, the consequence of this ability landing is to directly apply the Frightened condition to its target, instead of applying an effect that causes the Frightened condition. We seemingly fall into the second kind of wording, which means that you would not be afflicted by Frightened, and it would be safe to leave the paladin's aura.
Now, there could be the argument that this Frightened effect is a bit special, because of this part of the decription :
A creature can repeat the saving throw at the end of each of its turns, ending the effect on itselfon a success.
Since this Frightened effect doesn't behave like the regular Frightened condition, an possible interpretation could be that it is a different effect than the regular Frightened effect, and thus wouldn't be covered by the aura.
My interpretation is that this effect is not named differently. You are not afflicted by the effect called "Frightful Presence", but by the effect called "Frightened" with the extra that it can be ended early with a specific saving throw. In that sense, it should still count as an effect you cannot have while under the aura's protection.
Here are some examples of spells that can apply Frightened, and how to treat them according to the previous statements :
You project a phantasmal image of a creature's worst fears. Each creature in a 30-foot cone must succeed on a Wisdom saving throw or drop whatever it is holding and become frightened for the duration.
While frightened by this spell, a creature must take the Dash action and move away from you by the safest available route on each of its turns, unless there is nowhere to move. If the creature ends its turn in a location where it doesn't have line of sight to you, the creature can make a Wisdom saving throw. On a successful save, the spell ends for that creature.
Here, we fall into the case where a spell applies Frightened for its duration. Even if we nullify the Frightened effect applied by the spell, the target stays under the effect of of the Fear spell, as can be seen by this piece of text :
On a successful save, the spell ends for that creature.
This text implies that for the whole duration, the creature is under the spell's effect, which is the source of the Frightened condition. Removing the consequence of the spell's effect (the Frightened condition) doesn't remove its source (the Fear spell's effect).
The part we're concerned about is the following :
The enchantment causes creatures of the kind you designated to feel an intense urge to leave the area and avoid the target. When such a creature can see the target or comes within 60 feet of it, the creature must succeed on a Wisdom saving throw or become frightened. The creature remains frightened while it can see the target or is within 60 feet of it. While frightened by the target, the creature must use its movement to move to the nearest safe spot from which it can't see the target. If the creature moves more than 60 feet from the target and can't see it, the creature is no longer frightened, but the creature becomes frightened again if it regains sight of the target or moves within 60 feet of it.
Same thing here, although it's much easier to notice. The target is under the effect of the Antipathy/Sympathy spell, which will, under certain conditions, apply the Frightened condition. Being immune to this condition does not remove the spell's effect, meaning that leaving the paladin's aura will allow the spell's effect to apply the Frightened effect correctly.
The "fear" variant of the glyph has the following :
Fear. Each target must make a Wisdom saving throw and becomes frightened for 1 minute on a failed save. While frightened, the target drops whatever it is holding and must move at least 30 feet away from the glyph on each of its turns, if able.
This falls into the second case. The glyph inflicts the Frightened condition directly. The aura will prevent the affliction, meaning leaving the aura after the glyph triggered is safe.
Drawing on the deepest fears of a group of creatures, you create illusory creatures in their minds, visible only to them. Each creature in a 30-foot-radius sphere centered on a point of your choice within range must make a Wisdom saving throw. On a failed save, a creature becomes frightened for the duration. The illusion calls on the creature's deepest fears, manifesting its worst nightmares as an implacable threat. At the end of each of the frightened creature's turns, it must succeed on a Wisdom saving throw or take 4d10 psychic damage. On a successful save, the spell ends for that creature.
First case once more. The target is under the spell's effect, which causes multiple afflictions as long as the spell's effect is up, such as the Frightened condition. Being in the paladin's aura will prevent the Frightened condition as long as you stay in it. Whether the target takes the psychic damage or not is probably worth its own question.
For the spell's duration, your eyes become an inky void imbued with dread power. One creature of your choice within 60 feet of you that you can see must succeed on a Wisdom saving throw or be affected by one of the following effects of your choice for the duration. On each of your turns until the spell ends, you can use your action to target another creature but can't target a creature again if it has succeeded on a saving throw against this casting of eyebite.
The effect we're looking at is the following :
Panicked. The target is frightened of you. On each of its turns, the frightened creature must take the Dash action and move away from you by the safest and shortest available route, unless there is nowhere to move. If the target moves to a place at least 60 feet away from you where it can no longer see you, this effect ends.
Second case here. The spell's effect targets the caster and gives them the ability to use their action in order to inflict the Frightened condition on a target creature. If this affliction fails, then it will be safe to leave the paladin's aura (although the caster could simply use their action to try it again if you failed your saving throw).
You tap into the nightmares of a creature you can see within range and create an illusory manifestation of its deepest fears, visible only to that creature. The target must make a Wisdom saving throw. On a failed save, the target becomes frightened for the duration. At the end of each of the target's turns before the spell ends, the target must succeed on a Wisdom saving throw or take 4d10 psychic damage. On a successful save, the spell ends.
Almost identical to Weird, same interpretation.