Mesma aplicação, bancos de dados diferentes:Entity framework 6.X + MySQL + SQL Server

Então, a solução final é:

Crie seu próprio sucessor DbConfiguration com blackjack e prostitutas:

    public class MultipleDbConfiguration : DbConfiguration
    {
        #region Constructors 

        public MultipleDbConfiguration()
        {
            SetProviderServices(MySqlProviderInvariantName.ProviderName, new MySqlProviderServices());
        }

        #endregion Constructors

        #region Public methods 

        public static DbConnection GetMySqlConnection(string connectionString)
        {
            var connectionFactory = new MySqlConnectionFactory();

            return connectionFactory.CreateConnection(connectionString);
        }

        #endregion Public methods
    }

Marque Ms_SqlContext com MultipleDbConfiguration (e não faça mais nada com esse tipo de DbContext)

    [DbConfigurationType(typeof(MultipleDbConfiguration))]
    partial class Ms_SqlContext
    {
    }

Marque Ms_SqlContext com MultipleDbConfiguration e ajuste MY_SqlContext(string nameOrConnectionString) com chamada MultipleDbConfiguration.GetMySqlConnection(nameOrConnectionString)

    [DbConfigurationType(typeof(MultipleDbConfiguration))]
    partial class MY_SqlContext : DbContext
    {
                public MY_SqlContext(string nameOrConnectionString) : base(MultipleDbConfiguration.GetMySqlConnection(nameOrConnectionString), true)
                {}
    }

É ISSO!!!