Why does the local part move for both players when using touch function?

The touch event fires whenever any player touches the part, even if it's in a local script.

Though, I see what you are trying to do, to get the part to only move when the local player touches it.

To prevent the part from moving whenever any player touches it, you might want to check if the player that touched the part is the local one. If so, then do the loop.

local players = game:GetService("Players")
local player = players.LocalPlayer

Move = game.Workspace.MovingPlatform.Platform
Move2 = game.Workspace.MovingPlatform.Touch

Move2.Touched:Connect(function(hit)

    local humanoid = hit.Parent:FindFirstChild("Humanoid") -- Checking for a humanoid

    if(humanoid) then
        local character = humanoid.Parent
        local plr = players:GetPlayerFromCharacter(character) -- getting the player out of the character model

        if(plr == player) then
            -- Is the local player
            -- Keeping your code
            for i = 1, 100, 1 do
                Move.CFrame = Move.CFrame * CFrame.new(Vector3.new(0.1,0,0))
                wait() 
            end
        end
    end
end)

I would also suggest to have a look at the TweenService, it can come on pretty handy sometimes!