1: Make two contexts: one of users, another of system tables

I think the idea is to have all contexts equal, with only Connection String different. In this case, it is interesting to have a common context:

public class MeuContexto : DbContext 
{
    public MeuContexto(String connectionString) : base(connectionString) 
    { }
    // Coloque seus DbSets aqui
}

And another context that stores only user data:

public class UsersContext : DbContext 
{
    public MeuContexto() : base("DefaultConnection") 
    { }
    // Coloque seus DbSets de usuário aqui
}

In this user context, you can have a Model that stores the Connection Strings .

2: Implement a ActionFilter that determines which Connection String will be used

public class CustomActionFilter : ActionFilterAttribute, IActionFilter
{
    void IActionFilter.OnActionExecuting(ActionExecutingContext filterContext)
    {
        var usersContext = new usersContext();

        var user = usersContext.Users.SingleOrDefault(u => u.Name == User.Identity.Name);

        filterContext.HttpContext.Items["connectionString"] = user.Company.ConnectionString;

        this.OnActionExecuting(filterContext);
    }
}

3. Retrieve Connection String within your Action

In this case, you will have to start a context for each Action .

[CustomActionFilter]
public class MeuController : Controller
{
    public ActionResult Index()
    {
        using (var context = new MeuContexto(HttpContext.Items["connectionString"].ToString()))
        {
            // Use seu contexto aqui normalmente
        }
    }
}