The issue here is muddled by the other type changes that occured simultaneously upon the introduction of the fairy type.
The Fairy Type...
...takes double damage from...
...Poison-Type Attacks
...Steel-Type Attacks.
The significance of these match-ups is two fold. Steel prior to Gen VI was GREAT defensively but rarely used offensively. Who REALLY used Steel-Type Attacks? Adamant Serene Grace Jirachi's Iron Head, any user of Bullet Punch, uhh....yeah. Kind of about it. Steel only hit two things super-effectively: Ice and Rock. Both of which are hit by more common attacks that also hit a wider variety of targets (Flamethrower, Earthquake, Surf, Stone Edge...) It was for all intents and purposes only used for emergency coverage on things that would otherwise not have any other options (say Pidgeot with Steel Wing) or for STAB and Priority. By introducing a bunch of fairy-types that are really good for fighting, it only diversifies both the attacks used and the team combinations used.
On top of this, it gives people a reason to use Poison-type attacks. Let's be honest. The number of non-STAB Poison type attacks BEFORE Gen VI is a very short list consisting of Toxic, Toxic Spikes, and maybe the rare Venoshock. That's it. NO one used Poison type attacks because 1) Steel-types who are immune to poison were running around EVERYWHERE because of Stealth Rock and 2)Poison only hits Grass super-effectively and is resisted by GROUND, GHOST and ROCK (also Poison itself) which include a number of extremely powerful and useful Pokemon ranging from Gliscor to Gengar. Including a Poison-move was simply a bad choice in terms of oppurtunity cost.
By giving both Steel and Poison attacks SOME meaning in terms of offensive power, it allows a lot more Pokemon to have meaningful STAB attacks.
The fairy type...
...is resisted by...
...Poison type pokemon.
...Steel type pokemon.
....Fire type Pokemon.
Okay. I'll admit it. Steel resisting Fairy is not really super-important...or it wouldn't have been. But Steel's nerf to no longer resist Ghost and Dark means that it's Fairy-resistance is more important now. It's a net -1 defensively in terms of type resists but the fact remains that it's an important distinction that the Steel-Type is the defacto fairy killer.
Poison is also the rival fairy-killing type and is quite nice defensively actually: it has only two weaknesses to Ground and Psychic and a rather nice five resists including Fighting and Bug. The fact that is more relevant defensively means that Ground and Psychic are more relevant OFFENSIVELY (not that Ground needed it). While Psychic does see some use offensively, it's not as common as Fire, Ice, Ground, Rock, Water, or Electric.
Fire picking up another defensive resist is important because of Stealth Rock.
No. I'm serious. Remember how Charizard in DPP OU fell off the face of the earth because no one wanted to use Bellyzard? I still contend that Stealth Rock was THE most over-centralizing gameplay element in existence in recent memory, but the fact remains is that SR is really useful for checking Fire-types and giving people a reason to use Fire types is an important if confusing interaction.
The Fairy-type...
...is immune to...
Dragon-Type Moves.
The important thing here to note is not that the dragons have ways of "hitting around" the Fairy-type. It's that Choice-locked Dragon type attacks and/or Outrage can't just be spammed. You don't get to just send in Salamence and click Outrage to win once the Steels are gone. There's a whole bunch of people that that strategy is completely and utterly ignored. It encourages smarter play and it has a very weird if interesting fluff explanation for why it happens.
The Fairy-type...
...hits super-effectively on...
...the Fighting type.
...the Dark Type.
...The Dragon Type.
I have never understood why Fairy beats Fighting from anything except a gameplay standpoint. It's to nerf Fighting. It's so that Conkeldurr, and Machamp, and Infernape, and Blaziken, and Lucario Hitmontop, and Terrakion have to think twice. I don't understand HOW it works. I thought Psychic was the "magic" type (you know, like Quadratic Wizards vs. Linear Warriors?) so I'm not sure what's happening here.
I understand why it hits Dark super-effectively. It also nerfs the like twelve hundred (read: 30?) Pursuit users running around who just bashing on Ghosts and Psychics for the lulz. Think about it. This is way too cookie-cutter.
Vanilla Team A:
Stealth Rocker/Spiker
Spinblocker
Revenge Killer
Vanilla Team B:
Spinner
Pursuit-Trapper
SR-weak Sweeper
This also works alongside the buff that Dark got: it hits Steel neutrally now, but it is weak to Fairy now. So it's about a net +0.
Being super-effective to Dragon is note-worthy because besides other Dragons, the only thing that beat Dragons was Ice (which is admittedly super-common). This opens up some better options for Pokemon like Azumarill but this is mostly to nerf Dragons harder. Dragon is weaker offensively and defensively this generation and that is the important take-away.
So to answer the OP directly: I don't think that the Fairy-Type is OP, I think it has accomplished exactly what it needed to do, and that was shake up the status quo and rebalance the type match-ups so they people don't use Steels as much defensively and Dragons as much offensively.