How do I make a damage script that does not hit the player twice?

You're attempting to compare an object with an array! You would have to iterate through hitplayers and compare each index.. but rather than inserting the object as the value, insert it as the key for ease of access! I'll show you:

local hitplayers = {};

script.Parent.Touched:Connect(function(hit)
    local obj = hit.Parent
    local hum = hit.Parent:FindFirstChild("Humanoid")
    --separated conditions so it looks cleaner
    local outrider = hit.Parent.Name == "Outrider";
    local noWield = hit.Parent ~= script.Wielder.Value;
    local alreadyHit = hitplayers[obj]; --ease of access
    if outrider and noWield and not alreadyHit then
        hitplayers[obj] = true; --set as key
        hum:TakeDamage(50)
    end
end)

local DamagedPlayers = {}

function checkDamagedPlayers(PlayerName)
    for i,v in pairs(DamagedPlayers) do
        if v == PlayerName then
            return true
        end
    end

    return false
end

function onTouched(Part)
    local Character = Part:FindFirstAncestorWhichIsA("Model")


    if Character and Character.Humanoid then
        local inDamagedPlayers = checkDamagedPlayers(Character.Name)

        if Character.Name == "Outrider" and Character ~= script.Wielder.Value and inDamagedPlayers ~= true then
            table.insert(DamagedPlayers,Character)
            Character.Humanoid:TakeDamage(50)
        end
    end
end 

script.Parent.Touched(onTouched)

We have declare a table called "DamagedPlayers", and a function that checks if a specific item is within the table. The damage code is the same except we use the function checkDamagedPlayers.