UWP Combobox binding to SelectedItem property

As @Mike Eason and @kubakista said, you need set Mode explicitly. But this won't fix your issue completely.

In your code, your SelectedComboBoxOption is a string, but the SelectedItem is a ComboBoxItem object. Binding a String to SelectedItem won't change ComboBox's selected item. So if you want to use SelectedComboBoxOption to get and set ComboBox's selected item, you need to change SelectedComboBoxOption to ComboBoxItem and use a Convert in {x:Bind} to convert between Object and ComboBoxItem.

The Convert may like:

public class ComboBoxItemConvert : IValueConverter
{
    public object Convert(object value, Type targetType, object parameter, string language)
    {
        return value;
    }

    public object ConvertBack(object value, Type targetType, object parameter, string language)
    {
        return value as MainPage.ComboBoxItem;
    }
}

The XAML may like:

<Page ...>
    <Page.Resources>
        <local:ComboBoxItemConvert x:Key="ComboBoxItemConvert" />
    </Page.Resources>

    <Grid Background="{ThemeResource ApplicationPageBackgroundThemeBrush}">
        <StackPanel>
            <ComboBox Name="ComboBox"
                      DisplayMemberPath="ComboBoxHumanReadableOption"
                      Header="ComboBox"
                      ItemsSource="{x:Bind ComboBoxOptions}"
                      SelectedItem="{x:Bind SelectedComboBoxOption, Mode=TwoWay, Converter={StaticResource ComboBoxItemConvert}}"
                      SelectedValuePath="ComboBoxOption" />
            <TextBlock Name="BoundTextblock" Text="{x:Bind SelectedComboBoxOption.ComboBoxHumanReadableOption, Mode=OneWay}" />
        </StackPanel>
    </Grid>
</Page>

In the code-behind:

public sealed partial class MainPage : Page, INotifyPropertyChanged
{
    private ObservableCollection<ComboBoxItem> ComboBoxOptions;

    public MainPage()
    {
        this.InitializeComponent();
        ComboBoxOptions = new ObservableCollection<ComboBoxItem>();
        ComboBoxOptionsManager.GetComboBoxList(ComboBoxOptions);
        SelectedComboBoxOption = ComboBoxOptions[0];
    }
    ...
    private ComboBoxItem _SelectedComboBoxOption;

    public ComboBoxItem SelectedComboBoxOption
    {
        get
        {
            return _SelectedComboBoxOption;
        }
        set
        {
            if (_SelectedComboBoxOption != value)
            {
                _SelectedComboBoxOption = value;
                RaisePropertyChanged("SelectedComboBoxOption");
            }
        }
    }
    ...
}

If you just want to show selected item in TextBlock, there is an easy way. We can bind TextBlock's Text property to ComboBox's SelectedItem. And please note that SelectedItem's type is System.Object and {x:Bind} is strongly typed, and will resolve the type of each step in a path. If the type returned doesn’t have the member, it will fail at compile time. So we need to specify a cast to tell binding the real type of the object. But there is an issue while cast nested class in {x:Bind}. We can put ComboBoxItem out of MainPage as a workaround.

namespace ComboBoxTest
{
    public class ComboBoxItem
    {
        public string ComboBoxOption { get; set; }
        public string ComboBoxHumanReadableOption { get; set; }
    }

    public sealed partial class MainPage : Page, INotifyPropertyChanged
    {
        ...
    }
}

And in the XAML:

<Grid Background="{ThemeResource ApplicationPageBackgroundThemeBrush}">
    <StackPanel>
        <ComboBox Name="ComboBox"
                  DisplayMemberPath="ComboBoxHumanReadableOption"
                  Header="ComboBox"
                  ItemsSource="{x:Bind ComboBoxOptions}"
                  SelectedValuePath="ComboBoxOption" />
        <TextBlock Name="BoundTextblock" Text="{x:Bind ComboBox.SelectedItem.(local:ComboBoxItem.ComboBoxHumanReadableOption), Mode=OneWay}" />
    </StackPanel>
</Grid>