I think it should have been worded differently. I think it should be worded as follows:
Once per turn, when you hit a creature with a weapon attack roll, you can attempt to frighten that target. The target has disadvantage on its next attack and must make a Wisdom saving throw (DC equals 8 + your proficiency bonus + the ability modifier of the score increased by this feat). On a failed save, the target is also frightened of you until the end of your next turn.
I think the designers just thought that frightened causes disadvantage and just decided not to worry about it if the target failed the save and did not think through the mechanics. That is the only way I can reason through the ability and what it causes.
The example of someone that can't be frightened is probably the most common "wierd" effect, but it is not the only one.
On the flip side RAW if you can't be seen and use this against a target that can be frightened and it fails its save it is going to have no effect. So if you are say invisible and you use demoralizing strike the target is technically "frightened" but has no disadvantage if he fails his save ..... but if he makes his save he has disadvantage!