What i personally think is more significant than that dragon immunity is the resistances to both dark/fighting at the same time, which makes dark/fighting a lot less of a threatening coverage combo. For example, Hydreigon gets walled by most of the major fairy types, since they can take absorb Draco Meteor, Dark Pulse and Focus Blast (and lol U-turn). Mawile takes Earth Power and Flamethrower super effective, but Togekiss and Azumarril both are basically immune to the coverage move the other one is neutral to (not that I'm suggesting you should use both or all three in a team...).
Because most fairies are specially defensive rather than physically defensive, I think this is going to have more of an effect on special coverage, though not too much. Most stuff will just opt for shadow ball over dark pulse unless they get STAB, since ghost gets the same super effective hits while having only one immunity rather than three resistances. On the physical side, dark/fighting is probably more common, but fighting's usually on the stab side with considerably more base power tham you'll get specially, so I'm not sure what the effect will be. Azumarrill probably gets the best benefits out of the main fairies of being a physical fighting type counter, since the only coverage moves that'll hit it are uncommon stuff like Thunderpunch and Poison Jab, and it has great HP to make up for average defense.
This is getting me thinking - how are the most common fighting types gonna be affected by fairies? Mawile is obviously less of a threat, but I could see Azumarril and possibly Togekiss becoming pretty good fighting counters/checks.
Terrakion - Not sure how much damage prior he needs to die from aqua jet from Azumarril, but it's not much. Unless Togekiss can somehow Serene Grace make Stone Edge miss he's pretty dead though.
Lucario - He has steel STAB, but it's bullet punch which is usually a worse option than extremespeed outside of mega sets. I think though if he's lacking bullet punch or ice punch Togekiss straight up counters. Azumarril I'm less sure - haven't done any calculations here - but I think it depends on how well he can sponge a close combat and how much Aqua Jet does to -1 Luacrio.
Conkeldurr - Togekiss just laughs at this guy, though I think a stone edge on the switch can screw Togekiss back. I think Azumarril might have some trouble with bulk up sets with a little momentum behind them, but otherwise Conk does not like fairies.
Machamp - They both hate Dynamicpunch confusion, but otherwise both Togekiss and Azumarill pretty much straight out counter him.
Blaziken - I feel like Azumarill was basically made to counter this guy. Togekiss does a less impressive job, though.
Breloom - Opposite of the above.
Toxicroak - Toxicroak does an impressive job of absorbing Azumarril's waterfalls and being the only other fighting type than Lucario with supereffective stab to hit back with. I think Fighting/Dark/Poison even hits everything but himself neutral. Togekiss though can come in on not Poison Jab pretty well and screw Toxicroak 4 times over.
Infernape - Actually can kill all the fairies pretty well, since he's got really good coverage, but I think Azumarril will still beat his standard sets.
Heracross - Really hates fairies resisting both of his STABs, hates even more that his mega form can't outspeed togekiss as consistently anymore. At least it gets Skill Link Rock Blast to hit him on the with switch.
Medicham - Mega Medicham just shows how Mega Gardevoir is a lost opportunity, because even though Gardevoir can take most of what MegaCham throws at it typewise there is no way it's going to survive long enough.
Scrafty - poor guy...