A fire elemental seems to be (a creature made of) magical fire
Elementals (in general) are described on page 123 of the Monster Manual:
On its home plane, an elemental is a bodiless life force. Its dim consciousness manifests as a physical shape only when focused by the power of magic.
This would seem to indicate (to me, at least) that there is some sort of magic that is forming the flames into a cohesive, sentient creature and that this magic would counteract the Ice Troll's ability to extinguish non magic flames.
For what it's worth, the wording for Wild Shape reads:
Starting at 2nd level, you can use your action to magically assume the shape of a beast that you have seen before
A Circle of the Moon druid later gets to use this feature to turn into elementals instead, but the same "magically assume" phrasing would apply.
So the druid is magically changing into magically formed elemental which means that, more than likely, the druid in fire elemental form would not be extinguished by the Ice Troll's aura (though it would still take the cold damage since it is a creature).
Any fire that is started by coming into contact with the fire elemental is not described as magical however. The Fire Form provides one way of dousing the flames:
The first time it enters a creature's space on a turn, that creature [...] catches fire [...] until someone takes an action to douse the fire
If the target [hit by the Fire Elemental's Touch Attack] is a creature or a
flammable object, it ignites. Until a creature takes an action to douse the fire [...]
However, the Troll has it's own specific ability that allows it to extinguish non magical fire "automatically." This provides it with a second means of extinguishing the fire and it is not limited to the Elemental's dousing action.