The reason is because Aragorn was raised by Elrond but that's not quite as satisfying an explanation in and of itself.
This is also not exactly a thing in the books as others have said. Although for the films, it makes sense they would put this kind of emphasis on Elrond and Isildur's relationship for dramatic effect.
If you came into the films already knowing the books, I think it's quite a successful addition and plays through beautifully out to the end of the Return of the King when Elrond shows up to give his blessing to him and Arwen's marriage, if you're were not familiar with the books before seeing the films it places the important parts of the pre-history before the LOTR into a context that makes good sense.
To help you understand a bit more of that I'll give you the important background information that is not in the films, most of which in found in The Silmarillion.
It's essential to the story to realize that Elrond himself is Half-Elven and that he also had a twin brother named Elros. Because they are half-elves they were given the choice to choose immortality among the elves or to choose to live as mortal men.
In Elrond's case he choose to live as an elf and his twin brother Elros chose to live as a man and he went on to become the first King of Men in their ancient Kingdom of Númenor, which was destroyed thousands of years before The Lord of the Rings take place.
I'm not going to get into why that kingdom was destroyed but TL;DR Elendil was the father of Isildur and the last of the Lords of Númenor who had claim to the throne. He escaped the destruction of Númenor and sailed to Middle-earth where he founded the Kingdoms of Arnor and Gondor. And the rest you already know.
So Aragorn was not just raised like an orphan or something under Elrond's care, Aragorn is the last heir of a line of Kings that stretches back many thousands of years all the way back to Elrond's own twin brother. So this is something that is both very personal to him and that has weighed heavily on him for thousands of years.
Again, there are some things that were added for dramatic effect in the films, such as Elrond having to be convinced that victory would be achieved over Sauron and he had to be pushed by his daughter to do it who herself chose to live as a mortal in defiance of him.
But the gist of this part of story both in the films and the books is that due in no small part to that history, Elrond raised Aragorn as if he were his own son and did everything he could to assure he would fulfil his destiny and claim the mantle of King.