As Jason said, you shoud bind the Value property for the RadioButton. Such as:
<BindableLayout.ItemTemplate><DataTemplate><RadioButton x:Name="radio" Margin="0,0,15,0" Content="{Binding .}" Value="{Binding .}" GroupName="MenuCategories">
And then the selected value will show correctly. For more information, you can refer to the official document about Responding to a property change.