On Wiki it state that "Archery"
Archery is the art, practice, or skill of propelling arrows with the use of a bow, from Latin arcus.
While "archer"
A person who participates in archery is typically known as an "archer" or a "bowman"
Gilgamesh didn't use any bow or arrow at all, he basically just throw lots of sword or spear. (Wouldn't this make him a sword thrower?)
This is Archer from Fate/Zero
And this is Archer from Fate/Stay Night
Gilgamesh could possibly be fit in any class mainly thanks to all the items he has in the Gate of Babylon. He was quite possible chosen by the Holy Grail to fill the Archer Class as his personality would fit a very high level of Independent Action, the Wikia page also says:
He is able to easily ignore his Master's orders, making it possible to stroll around without making his Master aware of his whereabouts. The only way to command him is through the use of Command Spells, but his disposition that does not even contain an ounce of respect for his Master makes utilizing them idly a bad move.
It should be noted that the Servant Classes except the special classes (like Ruler, Avenger, Funny Vamp) allows a hero to be classified under more than 1 if they display the potential to be summoned as a different class. With the Archer Class, Heracles and Lu Bu were both summoned as Berserk but could be summoned as Archer, while Karna was summoned as a Lancer
Another example is Archer in the 5th Holy Grail War. The weapon he mainly uses in fact a pair of swords Kanshou and Bakuya, which we see him use for the first time during the first night as he fought Lancer. This annoys Lancer because he figured out Archer's Class but he was skilled with sword. Archer’s Bow is mainly just a normal Bow he projects and uses weapons like Caladbolg II as ammunition.
Though with that said, after the 3rd Holy Grail War and Avenger's corruption of the Great Holy Grail - the main engine to the ritual, the system for assigning classes to heroes may have been damaged. Both the 4th and 5th War's Archers didn't clearly fit the Archer Class, and normally only the Hassan-i-Sabbah should have been summoned as Assassin in the Fuyuki System, however Caster was able to summon Sasaki Kojiro.
You're being too literal. Archer as a label for a Servant class does not mean "This guy was badass with bows". It means "this guy is versatile and has potent long distance attacks." Several of the other Servant classes aren't particularly literal, either.
Saber just means they have potent close range melee weapons (usually swords) and great skill with them. None of the Sabers actually uses a Saber. They use longswords, greatswords, bastard swords, etc. A saber is a curved blade used by cavalry.
Lancer is potency in melee weapons with reach. None of the Lancers actually uses a lance. Mostly they use spears. And don't have horses with which to use a lance.
Casters are not required to literally cast anything. They simply have great facility in some fashion of magic. 5th war caster casts; 4th war caster uses a magical tome to summon; Cu Chulainn could have been a Caster thanks to his runes; Apocrypha Caster of Black fashions golems exclusively; Apocrypha Caster of Red writes stories in a book which makes them come true...
Assassin is normally very literal, though by system fiat rather than anything else. The grail was designed to only ever summon a Hassan of the Mountain, the head of the prototypical assassin's organization. But with sufficient manipulation of the grail anyone capable of stealthy or fast kills can be an Assassin. Fake Assassin is a Japanese Swordsman, summoned by a Caster whose magical talents were so great she could bend the rules of the grail to her will. In Apocrypha this restriction was lifted entirely. Apocrypha Assassin of Black was a serial killer that could summon an obscuring (and deadly) fog. Apocrypha Assassin of Red was typed as an Assassin because her history includes poisoning powerful people (and is a very bizarre Servant, in the technical sense of what the system usually does, for other reasons).
Rider need not have actually ridden the mounts they get as a Servant. They just need a strong connection to something that could be ridden. 5th War Rider's legend doesn't normally include her actually riding Pegasus, just that it was spawned from her corpse's neck. It was Bellerophon and Perseus whose legends are associated with actually riding the Pegasus. But that's her mount all the same (and is summoned by stabbing herself in the neck). And she had a skill that let her ride most anything short of dragons, despite an absence of riding anything at all in life. Riders also don't have to have animals involved in what qualifies them for "Rider". One of the PSP games let a ship captain qualify as a Rider, with their ship as the "mount".
Berserker's reasonably literal in the sense that what you get pretty solidly embodies the emotion of "berserk". But it's not literally the Nordic berserker the word comes from. And it's also a fairly simple requirement: the hero must have "gone berserk" at least once in life. Lots of great heroes are said to have flown into a fit of rage over something or another; often the death of a loved one. Hercules qualified because the Gods forcefully afflicted him with an insane, murderous rage at one point, not because he usually fought with near mindless rage and passion. Hercules was normally considered a very intelligent and cunning warrior. At various points in the FSN LN it is remarked that Berserker is probably Hercules' worst class to be summoned as, as he loses access to most of his greatest Noble Phantasms and abilities, and he probably would have been at his most powerful as an Archer. One can also intentionally summon a hero as a Berserker (as was done in both the 4th and 5th wars), even if this moment of rage was a fairly minor part of their history.
