What is the history of Aragorn?

I can think of two different reasons why Aragorn was called an Exile:

A King without a Kingdom

Aragorn's forebear Elendil founded the Kingdom of Arnor in the North. Arnor was the sister realm of Gondor in the South. When Elendil died, the Crown of Arnor passed to Isildur, his son.

Arnor eventually suffered from civil war and split up in smaller Kingdoms. All of the smaller Kingdoms were defeated by the Witch King. The Fall of Arnor left the House of Isildur without a Kingdom.

At the time of the War of the Ring, Aragorn was the heir of the House of Isildur. Because he was a King without a Kingdom, he may have been called an Exile.

An Exile from Númenor

The Kingdom of Númenor was an island raised by the One God as a reward for the faithful men who fought in the War against Morgoth. As years went on, the Kings of Númenor became ever more arrogant. In the end, they sent a large Armada against the Mayor Gods, hoping to wrest immortality from them.

The Mayor Gods called for help of the One God, and he destroyed the Armada and sunk the island of Númenor. Just before that, a few of the remaining faithful men had fled Númenor. Their leader was Elendil (the same as above.)

Because Aragorn was a descendant of Elendil, he could be said to be a Númenorean in Exile.