Because he walked across the land with his long legs, striding along.
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As Strider the Ranger, Aragorn is a wanderer and knowledgeable in the ways of nature. As King Elessar of Gondor, he resides in Minas Tirith, The White City.
Aside from Aragorn:-Estel ("hope"), in his childhood in Rivendell until he was told of his heritage.-Strider, by the people of Bree.-Elessar ("Elfstone") after the eponymous jewelled brooch given to him by Galadriel.-Elessar Telcontar, his ruling name. Consists of the aforementioned "Elfstone" which the people of Gondor called him, and the name Telcontar "Strider", which he added himself.-Thorongil ("Eagle of the Star"), during his time in the services of Thengel of Rohan and Ecthelion II of Gondor.-Dúnadan, which he is sometimes called in Rivendell and by Bilbo, who gave im that name in reference to his heritage as Chief of the Dúnedain.There are also various other praising or derogatory names that people called him by, such as "Longshanks" (by Bill Ferny) or "Wingfoot" (by Éomer) or "Envinyatar" ("renewer") but those are not proper names and/or were not used as names.
His real name is Aragorn II, but once he becomes the king of Gondor he is called Elessar Telcontar.
This is a scene in the Special Extended Edition and it is not of Arwen, it is of Aragorn's mother Gilraen. That is why the scene is called "Gilraen's Memorial".
The Tale of Aragorn and Arwen relates that Arwen had a son, Eldarion, and at least two unnamed daughters by Aragorn.