When I click the part, the Sword should be in my inventitory?

The Backpack is a child of the player, not inside their character. Try this:

local Detector = script.Parent.ClickDetector
local Tool = game.ServerStorage.ClassicSword
local Part = workspace.Purple

Detector.MouseClick:Connect(function(Player)
    local Clone = Tool:Clone()
    Clone.Parent = Player.Character
    Part:Destroy()
end)



workspace.Purple.Touched:Connect(function(hit)
    local player = game:GetService("Players"):GetPlayerFromCharacter(hit.Parent)
    if hit.Parent:FindFirstChild("Humanoid") and player then
        if player.Backpack:FindFirstChild("ClassicSword") or hit.Parent:FindFirstChild("ClassicSword") then
            print("There is a sword")
        end
    end
end)

It gets the player from its character (hit.Parent.)