public class NotificationObject : INotifyPropertyChanged
{
    public event PropertyChangedEventHandler PropertyChanged;

    protected virtual void OnPropertyChanged([CallerMemberName] string propertyName = null)
    {
        PropertyChanged?.Invoke(this, new PropertyChangedEventArgs(propertyName));
    }

    protected void SetField<T>(ref T field, T value, [CallerMemberName] string propertyName = null)
    {
        if (!EqualityComparer<T>.Default.Equals(field, value))
        {
            field = value;
            OnPropertyChanged(propertyName);
        }
    }
}

public partial class MainWindow : Window
{
    ModalViewModel mainView;

    public MainWindow()
    {
        InitializeComponent();
    }

    private void Button_Click(object sender, RoutedEventArgs e)
    {
        mainView = new ModalViewModel(new ModalWindow());
        mainView.SetMessage("Окошко!");
        mainView.ShowWindow();
    }
}

public class ModalViewModel : NotificationObject
{
    private ModalWindow _modalWindow;

    private string _message;
    public string Message
    {
        get => _message;
        set
        {
            _message = value;
            // Этот метод уже реализовывает OnPropertyChanged("Message");
            SetField(ref _message, value);
        }
    }

    public ModalViewModel()
    {
        _message = "Hi"; // Тут работает
    }

    public ModalViewModel(ModalWindow modalWindow)
    {
        _modalWindow = modalWindow;
        _message = "Hello!"; // Тут не работает
    }

    public void ShowWindow()
    {
        _modalWindow.Show();
    }

    public void SetMessage(string message)
    {
        _message = message;
    }
}

<Window x:Class="DataContextAndBindingTest.View.ModalWindow"
    xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
    xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
    xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
    xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
    xmlns:local="clr-namespace:DataContextAndBindingTest.ViewModel"
    mc:Ignorable="d"
    Title="ModalWindow" Height="300" Width="300">
<Window.Resources>
    <local:ModalViewModel x:Key="ModalViewModel" />
</Window.Resources>
<Grid DataContext="{StaticResource ModalViewModel}">
    <Label VerticalContentAlignment="Center" HorizontalContentAlignment="Center" Margin="0,24,0,85"
          FontSize="18" Content="{Binding Message}"/>
    <Button Height="30" Width="100" Content="Ok" Margin="96,208,96,31" />
</Grid>