public class IntTextBox : UserControl, INotifyPropertyChanged {

 	public int Value {
            	get { return (int)GetValue(ValueProperty); }
            	set { SetValue(ValueProperty, value); }
        }

        public static readonly DependencyProperty ValueProperty =  DependencyProperty.Register(
            "Value", typeof(int), typeof(IntTextBox), new PropertyMetadata(0, OnValueChangedCallBack));
            
       public event PropertyChangedEventHandler PropertyChanged;
          
        /*
          	Removed all the initialization code and the code that actually changes the value for simplicity
        */
            
  	protected static void OnValueChangedCallBack(DependencyObject sender, DependencyPropertyChangedEventArgs e) {
   		IntTextBox intTextBox = sender as IntTextBox;
            	intTextBox?.OnPropertyChanged("Value");
        }

        protected void OnPropertyChanged(string name) {
            	PropertyChangedEventHandler handler = PropertyChanged;
           	if (handler != null) {
                	handler(this, new PropertyChangedEventArgs(name));
            	}
        }
}

<MyControl:IntTextBox Value="{Binding Value}"/>

public class IntTextBoxViewScript : INotifyPropertyChanged {
    	public event PropertyChangedEventHandler PropertyChanged;
    	
    	private int _value;
    	public int Value { get { return _value; } set{ _value = value; OnPropertyChanged("Value"); }}
    	
}

...
<TextBlock Text="{Binding Value}"/>
...
<ContentPresenter ContentTemplate="{StaticResource IntBoxTemplate}" Content="{Binding IntBoxTemplateScriptObject}"/>

public class EngineConfigViewScript : MonoBehaviour {
	private IntTextBoxViewScript _intBoxTemplateScriptObject = new IntTextBoxViewScript();
    	public IntTextBoxViewScript IntBoxTemplateScriptObject{ get => _intBoxTemplateScriptObject; } 
    	
    	private void Start() {
        	NoesisView view = GetComponent<NoesisView>();
        	view.Content.DataContext = this;
    	}
}

Text="{Binding Value, Mode=TwoWay}"