Should I take all the swords?

Before setting out on an adventure, you can customize what you bring with you. After a bit, you eventually have multiple forms of weaponry (bone spear, iron sword, etc). Each has it's own weight so carrying multiples could end up being an issue.

Is there any benefit from taking one of each (1 sword, 1 spear) or multiples of the same (2 spears)?

Ranged weapons have obvious benefits since there are different types of ammunition, but for melee, it seems only the best weapon is used.

There is a benefit to taking one of each type of weapon. When attacking, they each have their own unique attack, and cooldown. You will do much more damage is you bring a sword and a spear, as you will get two attacks, a stab for the spear, and a slash for the sword. Doing this is actually one of the better ways to kill strong enemies, as they will hit you pretty hard, even with armor.

You can also use a Bola to stun your enemy, but be careful, as those are one-shot items.

Bring multiples of the same weapon isn't needed, and shouldn't be done; there is no benefit to doing so, and if you die, you lose everything you took with you.

You should also not bring more than one of each weapon "type" (except for consumables, like Bolas). For example, one Steel Sword, one Iron Sword, and one Bone Spear.

Keep in mind you must carry the weight of each additional weapon, lowering your carrying capacity for other loot.

Frank's answer is a good one, but there is also a secret involving the weapons you use. I don't recommend reading this until after your first playthrough:

Using unarmed attacks (i.e. holding no weapon at all) trains the character as an unarmed fighter; after attacking 50, 150, and 300 times you will earn the boxer, martial artist, and unarmed master perks, which make you better at punching.
The boxer perk will double the damage you do in combat (two damage per hit). martial artist will triple that again (six damage per hit). unarmed master will double that again (12 damage per hit) and cut the attack cooldown in half!
So it's up to you: would you rather carry multiple weapons and have multiple attacks, or have a single strong attack?