Why does it say that variable part is a non-existing Nil value?

Your part.Touched event is not triggering because of how you positioned it inside your script. Don't have it inside your Players.PlayerAdded event.

EDIT:

Your sound was not being copied to the PlayerGui because the server cannot access it due to how FilteringEnabled works. You would have to use RemoteEvents or RemoteFunctions for that. Lucky for you there's no need for that in your case: just manually put your sound inside StarterGui in studio.

However you will still need to have a RemoteEvent call FireClient() when your part gets pressed, then have a LocalScript play your sound:

Inside a regular script:

local part = game.Workspace.Music
local RemoteEvent = Instance.new("RemoteEvent", part)
canplay = true 

local function muziekspelen(otherpart)
	print(otherpart)
	local humanoid = otherpart.Parent:FindFirstChild("Humanoid")

	if humanoid and canplay then
		local player = game.Players:GetPlayerFromCharacter(otherpart.Parent)
		canplay = false
		RemoteEvent:FireClient(player)
	end

	canplay = true
end

part.Touched:connect(muziekspelen)

Inside a local script inside StarterGui:

local RemoteEvent = game.Workspace:WaitForChild("Music"):WaitForChild("RemoteEvent")
local player = game.Players.LocalPlayer

RemoteEvent.OnClientEvent:Connect(function()
	local sound = player.PlayerGui:WaitForChild("sound")
	sound:Play()
	sound.Looped = true
	print(otherpart)
end)