In Fairy Tail, there are many types of magic, not just elemental or dragon slayer magic. If you read/watch the arcs dealing with the history of the world and the Fairy Tale guild, you'll see Mavis and the founders/masters of the various guilds and many others discovering, inventing or developing new magic. In the various other arcs, you'll also see things like Natsu learning transformation magic or being able to absorb lightning, not just fire. So while someone may show affinity for a certain type of magic, that doesn't mean they can't do/learn others.
(BTW even in Naruto, all they had were affinities for certain elements. They could still learn techniques/jutsu from other elements, it was just easier to learn jutsu falling under their own affinities and less of a power drain to use. Kakashi even mentioned that while most shinobi only had one or two affinities, by the time you were a jounin you should be able to use jutsu from all the elements.)
However, if you need certain types of training to perform certain types of magic, you are not likely to suddenly be able to do it without that training. Also, in the Fairy Tale world, you could only get dragon slayer magic by training under a dragon of that element or artificially (like Laxus in the beginning) from a dragon crystal.