The Realms-In-Exile were founded by the Faithful Numenoreans who survived the fall of Numenor in 3320 Second Age (SA). After the War of the Last Alliance (3434-41 SA) the relative standing of each throne changed, with Gondor growing to become a mighty power, whereas Arnor split into three seperate kingdoms, Cardolan, Rhudaur and Arthedain and declined in strength until it was destroyed in war with Angmar.
In the Second Age Gondor was ruled cojointly by Elendil's sons Isildur and Anarion. Elendil ruled in the northern realm of Arnor and was High-King of Gondor and Arnor. The institution of the high-kingship is no where defined. It does not appear to have provided any authority in the affairs of the other kingdom(s). However, it conferred on a ruler a superior stature in dealings between the kingdoms, notably in wars involving the associated kingdoms. Hence, the leaders of the Last Alliance were Gil-Galad and Elendil, the High-Kings of their respective folk.
The ruling Dunedain of Arnor and Gondor were bound not only by the High-Kingship, but by kinship and a common foe, Sauron. Therefore, when Sauron attacked Gondor, Elendil formed the Last Alliance with Gil-Galad (3430 SA) and marched south with the Men of Arnor.
The war witnessed the overthrow of Sauron's power and large loss of life, including the death of Elendil and Anarion, and soon after, the death of Isildur at the Gladden Fields (2 TA). Menedil, born 3318 SA the fouth child of Anarion, became King of Gondor; Valandil, Isildur's surviving son, born c. 3431 SA, assumed the kingship of Arnor in 10 TA when he was around nineteen. The seniority of Menedil over Valandil and the gap of 8 years before Valandil assumed the throne probably contributed to a change in political relations. Though it is unstated, I think...