As we see in the series, all Dragon Slayers (not God Slayers) who are trained by Dragons have Exceeds as partners.
Specifically there are 5 of them: Natsu, Gajeel, Wendy, Sting and Rogue. Now, Laxus is also a Dragon Slayer, but he was not trained by a Dragon and became a Dragon Slayer by Lacrima. Is this a coincidence? Or is there a reason behind it?
If we follow the story of Fairy Tail from the beginning, I think it goes like this. At first, Happy was the only Exceed. Mashima-sensei created Happy to be some-kind of sidekick to Natsu. And then there's Gajeel. At that time Mashima-sensei still has no idea about the Edolas Arc, so Gajeel was on his own, even when he revealed that Laxus' thunder magic was in fact a dragon slayer magic. Then there's Wendy, another dragon slayer. Wendy is the first female dragon slayer, and since Happy was the only flying cat (the term Exceed was not revealed at that time), he decided to make another flying cat, named Charle to accompany her.
It was during this that Mashima-sensei got the idea to make Exceed as the official sidekick of Dragon Slayers. That's why he then make Gajeel wonder why he is the only Dragon Slayer without a cat. And after that, Exceed is the official sidekick of Dragon Slayers. So when he created Sting and Rogue, he created Frosche and Lector too.
Edit: The Exceed is sidekick of dragon slayers that was raised by dragons, that is, a real dragon slayer, not the one made by embedding lacrima such as Laxus (Thunder Dragon Slayer) and Cobra (Poison Dragon Slayer).
One-hundred Exceed eggs, including Happy and Carla, were taken from their parents and sent to Earth Land on a mission to obliterate the Dragon Slayers. When Faust began using Anima to steal Magic Power from Earth Land, Queen Shagotte gave the Exceed in the field new orders: To capture the Dragon Slayers and bring them to Edolas. The Dragon Slayers would then be used to create a special type of Lacrima filled with Dragon Magic.
Carla decides to abandon her mission to kill Wendy and brought her to Edolas without knowing that her mission had been changed to simply lure Wendy to Edolas. She becomes very distressed by the fact that she unknowingly betrayed Wendy and fulfilled her mission. Happy is unable to recall the details of his mission but refuses to be a puppet of the Exceed, making it clear that he and Carla are members of Fairy Tail first and Exceed second. After this, they escape and are declared outlaws.
This mission was later revealed to be a lie, due to the Queen's power she could see the events of the Edolas arc, and as such, she saw Extalia destroyed. So in order to keep the species alive, she sent 100 eggs to Earth Land to protect them, and lied about the mission to kill the Dragon Slayers to the other Exceed.
Source : http://fairytail.wikia.com/wiki/Exceed (specific revision)
There are no connections between Exceeds and Dragon Slayers. Hiro seems to always pair them up without any reason.
Remember, Gajeel did not want an exceed initially, but after seeing Wendy and Natsu with one he wondered why he didn't have one. He then became jealous of Natsu and Wendy and wanted Lilly as his comrade.
Second gen dragon slayer don't get exceeds because there not connected to dragons a lacrima was put inside them to give that magic
I think it's just a coincidence that all of the First Gen Dragon Slayers have Exceed partners. After all, we find out that the whole "Kill/Bring the Dragon Slayers to Edolas" thing was made up by the queen.
All the First and Third generation dragon slayers have exceeds because they are basically true dragon slayers because they are or were raised by a dragon. The queen of extalia which is now no more created a fake mission to capture and kill the dragon slayers so the future for the exceeds could be safe, including her daughter, Carla. The only exceed that later became a sidekick of an dragon slayer that didn't get sent to earthland is Panther Lily, which Gajeel made his sidekick because he was jealous of Natsu and Wendy with Happy and Carla.
I agree with Ayase Eri answer.
However i like to add some of my own points based on the episodes which shows other Exceeds after Edolas Arc. Mashima didn't have any original plan regarding to have Exceeds with Dragon slayer, he only created Happy as a sidekick for Natsu. Later he decided to assign Carla to Wendy so that he can create Edolas Arc. After Edolas Arc he decide to Make Exceed as official side kick for Dragon Slayer and made Lily as Gajeel partner, but i don't think there was any specific reason to assign Exceed with Dragon slayer other than creating Edolas Arc, because in later episodes Nichiya(Exceed) join Blue Pegasus guild which does not have any Dragon Slayer. Also i think people want to see an Exceed with Dragon slayer that why when he created Sting and Rogue, he created Frosche and Lector too.
