For the sake of completeness, I have decided to post my final answer. This curated answer is mostly a resume of this huge thread that focuses on the most relevant issues and explores other options I also found. I hope future readers find it interesting.

Why this happens

So, after a lot of investigation and help from the community I found out what is happening.

Turns out that session data (data that needs to travel between multiple pages) cannot be shared over websockets. LiveViews use Websockets and therefore suffer from this limitation (How to get app.html.heex to use assigns properly with LiveView - #18 by sodapopcan - Questions / Help - Elixir Programming Language Forum):

Yes, if you are logging in a user you have to store a successful login id in a session. The thing is, you can’t set session data over a websocket. This is a websocket limitation, not a LiveView one ( … )

This also means you cant use cookies/alter them (Persisting data across liveview navigation - #4 by soup - Questions / Help - Elixir Programming Language Forum):

( … ) because LV is all over a websocket you cant alter the cookies so your stuck with local storage or another system.

Possible solutions

However, this is not the end. There are other options if you need to share data between websockets. I was able to identify 4:

1. Add the session data to the URL of the target page. An example of this is using GET HTTTP method with session data as parameters (or perhaps using post): elixir - How to get app.html.heex to use assigns properly with LiveView - Stack Overflow
2. Save the data inside a global ETS table in the server: Persisting data across liveview navigation - #4 by soup - Questions / Help - Elixir Programming Language Forum
3. Save the data in an Agent. You could have an Agent per websocket, thus avoiding a global state and improving efficiency: Persisting data across liveview navigation - #4 by soup - Questions / Help - Elixir Programming Language Forum
4. Store the data in a Phoenix Session: How to get app.html.heex to use assigns properly with LiveView - #18 by sodapopcan - Questions / Help - Elixir Programming Language Forum

Here you have to weight the risks/benefits of every solution. If you have your own website, solution 4 would probably be the most secure, while solution 1 would also be OK provided you encrypt the data before putting it in the url parameters or post body request.

However, in my specific case, a Windows desktop application for a single user, these considerations are not important. I have therefore opted for solution 2, since it is simpler than solution 3 and the data there is mostly read only.

Final code

So now I am using a temporal persistence layer to store session information:

defmodule WebInterface.Persistence do
  @moduledoc """
  Responsible for temporary persistence in WebInterface. Uses ETS beneath the scenes.
  """

  alias ETS
  alias Shared.Data.User

  @table_name :data

  @spec init :: :ok | {:error, any}
  def init do
    with {:ok, _table} <- ETS.KeyValueSet.new(name: @table_name, protection: :public) do
      :ok
    end
  end

  @spec set_user(User.t) :: :ok | {:error, any}
  def set_user(%User{} = user) do
    with {:ok, table} <- ETS.KeyValueSet.wrap_existing(@table_name),
      {:ok, _updated_table} <- ETS.KeyValueSet.put(table, :user, user) do
        :ok
      end
  end

  @spec get_user :: {:ok, User.t} | {:error, any}
  def get_user do
    with {:ok, table} <- ETS.KeyValueSet.wrap_existing(@table_name) do
      ETS.KeyValueSet.get(table, :user)
    end
  end

  @spec has_user? :: boolean
  def has_user? do
    case get_user() do
      {:ok, nil} -> false
      {:ok, _user} -> true
      _ -> false
    end
  end
end

Now in my app file, I simply ask the persistence module if I have the data I want. This is quite transparent:

app.html.heex:

<header>
 <%= if WebInterface.Persistence.has_user?() do %>
  <h1> <%= @user.name %> is awesome !</h1>
  <% end %>

</header>
<main class="">
  <div class="mx-auto max-w-2xl">
    <.flash_group flash={@flash} />
    <%= @inner_content %>
  </div>
</main>

And I set/get the data I want via the Persistence API.

user_login_live.ex:

defmodule WebInterface.UserLoginLive do
  use WebInterface, :live_view

  alias Manager
  alias Shared.Data.{Credentials, User}
  alias WebInterface.Persistence

  @impl true
  def mount(_params, _session, socket), do: {:ok, socket}

  @impl true
  def handle_event("login", %{"email" => email, "password" => password} = params, socket) do
   # this is an async request for authentication
   # here we simply start it 
   :ok =
      email
      |> Credentials.new(password)
      |> Manager.login(Map.has_key?(params, "remember-me"))

      {:noreply, socket}
  end


  @impl true
  def handle_info({:login, %User{} = user, :done}, socket) do
    # when we receive this message, we know authentication is done.
     
    # and in other places we use get_user/0 to retrieve session data
    :ok = Persistence.set_user(user)

    {:noreply,  socket |> redirect(to: ~p"/other_page")}
  end
end

For now this solution works wonderfully.