elixir test fails with "Reason: already started"

I'm currently working on my first big elixir project and wanted to properly utilize testing this time. However, if I add my Modules to the "normal" supervisor, i cannot start them again with start_supervised! and all tests fail with Reason: already started: #PID<0.144.0>

Here is my code:

(application.ex)

    defmodule Websocks.Application do
  # See https://hexdocs.pm/elixir/Application.html
  # for more information on OTP Applications
  @moduledoc false

  use Application

  def start(_type, _args) do
    children = [
      {Websocks.PoolSupervisor, []},
      {Websocks.PoolHandler, %{}}
      # {Websocks.Worker, arg}
    ]

    # See https://hexdocs.pm/elixir/Supervisor.html
    # for other strategies and supported options
    opts = [strategy: :one_for_one, name: Websocks.Supervisor]
    Supervisor.start_link(children, opts)
  end
end

Some of my tests:

defmodule PoolHandlerTest do
  use ExUnit.Case, async: true
  alias Websocks.PoolHandler
  doctest PoolHandler

  setup do
    start_supervised!({PoolHandler, %{}})
    %{}
  end

  test "adding two pools and checking if they are there" do
    assert PoolHandler.add(:first) == :ok
    assert PoolHandler.add(:second) == :ok
    assert PoolHandler.get_pools() == {:ok,%{:first => nil, :second => nil}}
  end

and the pool handler:

defmodule Websocks.PoolHandler do
  use GenServer

  # Client
  def start_link(default) when is_map(default) do
    GenServer.start_link(__MODULE__, default, name: __MODULE__)
  end

  # Server (callbacks)

  @impl true
  def init(arg) do
    {:ok, arg}
  end
end

(I cut out the stuff i think is not necessary, but the complete code is on github here: github)

Thanks in advance for any help i get!