Until Isabella and Ferdinand, the major kingdoms of the Iberian Peninsula were Castile and the Crown of Aragon. As far as I know, Catalonia was always a territory whether of Aragon, France and now Spain. However, since 19th century is Catalonia, and not Aragon, the region that has developed a strong nationalist sentiment and nationalist political forces. When I look at the map of "Grand Catalonia" (països catalans) it contains just East Aragon, and the capital of the "Catalan nation" is supposed to be Barcelona, not Zaragoza. The nationalists name themselves "catalans" and not "aragonese". But Catalonia had never a king - Aragon did. Aragon was the actual sovereign kingdom, with their own overseas empire.
I'm aware that Catalonia it's been always somehow special and that they had their own Parliament, but I don't know how did they end up considering themselves a nation instead of claiming the spirit of the Crown of Aragon as a whole. How and why did this happen?
You appear to be confusing the Crown of Aragon, which was all the territories, with the Kingdom of Aragon, which was only a part of it. The reason why it is known as the Crown of Aragon is because at the time it was percieved that a "Kingdom" had more status than a "Principality" (as the "Principality of Catalonia" was known). However, that is about it: status in "words". The economic capital, where more people lived and were the kings of Aragon mostly resided was Barcelona, which is today the capital of the autonomous region of Catalonia. For instance, even though it was known as the Kingdom of Aragon, its inhabitants where often known as catalans, even though sometimes they were not (for instance, the Borgia family had Valencian origins, but were sometimes referred to as catalans).
Why today there is no strong Aragonese political movement is an interesting question. In my opinion, it has to do with a weaker economy and the disappearance of the aragonese language.