The intent is for creatures to make a save when the spell is cast
You are correct that a literal reading of the spell description would mean that targets are only outlined if they fail an unrelated Dexterity saving throw during faerie fire's duration. A search of D&D Beyond turned up no other spells with the same wording. As you point out, spells that require a saving throw are typically worded like burning hands:
Each creature in a 15-foot cone must make a Dexterity saving throw.
I believe that the omission of a saving throw for faerie fire is likely an oversight. The only alternative is that a common 1st-level spell has a unique spell mechanic that only really serves to give it an extremely niche use-case.
A literal leading of faerie fire also makes it nearly useless for non-spellcaster drow, despite the fact that it (and darkness) are staples of drow combat in official adventures and novels.
There is precedent for effects triggering on arbitrary saving throws
The star spawn larva mage has the following reaction (MToF, p. 235):
Feed on Weakness. When a creature within 20 feet of the larva mage fails a saving throw, the larva mage gains 10 temporary hit points.
The larva mage doesn't have to be the one to force a saving throw. In fact, the saving throw could be caused by a PC targeting one of the mage's allies. All that said, this is a reaction ability on a CR 16 creature that is explicitly connected to bizarre cosmic horrors. It is unlikely that faerie fire is intended to be the one other case of this strange mechanic.