From Did Gandalf know the origin of "Glamdring"? we learn that Glamdring belonged to the King of Gondolin (Turgon) and that Elrond suspects it was looted from the remnants of the city by orcs or dragons.
Rivendell was founded by Elrond and other survivors/descendants of Gondolin. Elrond's father Eärendil was born in Gondolin, and King Turgon was Elrond's great grand-father. Legally, Glamdring would actually belong to Elrond by right of heritage 1), Elrond being Turgon's closest living relative in Middle-Earth.
Also, the sword belonged to the former High King of Noldor, so having it in his possession would strengthen Elrond's position and claimed heritage from a political point of view. Much like Narsil did for Aragorn.
Yet Elrond lets Gandalf have the sword almost dismissively!
Sure, he might think that Gandalf has greater use for it, just like Cirdan thought Gandalf would have greater use of the ring Narya. But he doesn't even seem particularly surprised or even excited upon encountering an old heirloom of his own family, which has been lost for several millennia.
Compare this with with Elrond's fascination over the sword Narsil, and his bland reaction upon finding Glamdring seems even stranger.
And then Gondolin was located in the old world, supposedly somewhere vaguely far north of Rivendell. If the swords were originally looted by orcs/dragons, then carried around all over the world in various loot piles for two ages, isn't it a strange coincidence that they are found so close to Rivendell?
I am aware that these swords (Glamdring and Orcrist) are kind of an anachronism, as Tolkien borrowed bits and pieces from The Fall of Gondolin in his latter works. But perhaps there exists a canonical explanation, a 'patch' released by Tolkien afterwards, like in the case with the Glorfindel anachronism in LOTR?
1) This based on the nobility of all people in Middle-Earth using the same primogeniture inheritance (male preference) system as feudal Europe, regarding titles and royal regalia. Aragorn inherits Narsil from Isildur. Elwing inherits the Silmaril from Thingol, Turin inherits the dragon helm from Hurin, Beren inherits the ring of Barahir from Barahir. And so on. It is a recurring, undisputed rule of heritage throughout the books.