Neither. The Archer class servant of the 5th Fuyuki Grail War was a magus before anything else. At his core he is Projection and Reinforcement magecraft, all culminating into his reality marble (an innate bounded field) "Unlimited Blade Works."
As a magus his talents were unremarkable, save for his talent in Reinforcement and Projection. While Emiya Shirou was in his school's archery club, it was short-lived and not very noteworthy. Most of his strength came from him making a contract with the World in order to become a Heroic Spirit in the form of a Counter Guardian after his passing.
His adeptness in melee combat is unusual for a servant of his class, but his Origin is "Sword." It's inevitable that we would use them in one way or another. The twin swords he uses, Kansho and Bakuya are not tied to him as a servant, but was merely something he projected. He utilizes these twins swords because of the innate magic resistance (to deflect Caster's A-ranked spells) they give, in addition to being effective against monsters such as Gorgon form of Rider and the extradimensional abomination summoned by Caster (from Fate/Zero).
Servants are placed under the Archer class based on their Noble Phantasm. The user does not have to be adept at a bow, but any projectile weapons, even firearms. Gilgamesh is placed under Archer based on his Noble Phantasm "Gate of Babylon," which essentially is him launching Noble Phantasms stored in his treasury.
"Unlimited Blade Works" allows allows him to reproduce just about any weapon he's seen (with some exceptions), allowing him to summon an arsenal of weapons as he needs it. His Projection magecraft allows him to create and overload projected Noble Phantasms (the copies are one rank lower than the original). The overloaded Noble Phantasms called Broken Phantasms are increased by one, but are destroyed they are used. Combining Broken Phantasms with a bow allows Archer to nuke the enemy from extreme distances. Think of as having an unlimited quiver of super powerful arrows for just about any situation. Tactics such as these make him suitable to be classified as an Archer class servant.
Additionally, it should be noted that this Archer is not from the same timeline as Emiya Shirou. He is an Emiya Shirou that walked a different path as a nameless hero of justice, who made a contract with the world to be save humanity as a whole as Counter Guardian. But in the end, he was made to clean up after the actions humans by becoming an assassin. To save humanity as a whole, instead of "saving everyone" as he had hoped.
It's possible for him to be summoned as Caster, Berserker, or Assassin, but that's a story for another time.