The problem has nothing to do with the SpeechSynthetizer, but is related to the code you're using to call your method:

private void OnClick(object _sender, RoutedEventArgs _e)
{
    SpeakAsync("en-US", "Test 1").Wait();
    SpeakAsync("en-US", "Test 2").Wait();
}

It's using an async method with task Wait() from a UI thread which causes a deadlock.

This is a classic mistake (same as with Winforms or WPF), see this How to avoid WinForm freezing using Task.Wait or Winforms call to async method hangs up program on this site or Don't Block on Async Code for a more detailed answer.

So the solution is just to do this instead:

private async void OnClick(object _sender, RoutedEventArgs _e)
{
    await SpeakAsync("en-US", "Test 1");
    await SpeakAsync("en-US", "Test 2");
}

Note with existing code, both phrases will play together, here's a version that can optionally wait for the end of play:

private Task SpeakAsync(string language, string text, bool waitForEnd = true)
{
    if (waitForEnd)
    {
        var tsc = new TaskCompletionSource();
        Task.Run(() => Do(tsc));
        return tsc.Task;
    }
    return Do(null);

    async Task Do(TaskCompletionSource tsc)
    {
        var player = new MediaPlayer();
        using var synth = new SpeechSynthesizer();
        synth.Voice = SpeechSynthesizer.AllVoices.FirstOrDefault(v => v.Language == language) ?? SpeechSynthesizer.DefaultVoice;
        var source = await synth.SynthesizeTextToStreamAsync(text);
        player.MediaEnded += (sender, _) =>
        {
            source.Dispose();
            tsc?.SetResult();
        };
        player.SetStreamSource(source);
        player.Play();
    }
}