Improvment for your tests

I've added IGameClient for GameClient.
This is necessary so that you can write tests, like this

public static class TestExtensions
{
    private static readonly ObjectSerializer serializer = new();

    public static Task<ServerResponse<T>> AsSuccessResponse<T>(this T value) where T : IGameObject
    {
        var data = serializer.Encode(value).ToString();
        return Task.FromResult(new ServerResponse<T>(HttpStatusCode.OK, data));
    }
}

public class Tests
{
    public void MyFavoriteTest()
    {
        var gameClient = A.Fake<IGameClient>();
        A.CallTo(() => gameClient.LoginAsync("test", "123")).Returns(new LoginResponse()
        {
            AccountId = 111,
            UserId = 333
        }.AsSuccessResponse());
    }
}

A.Fake it is FakeItEasy library, see documentation if you are interested