UserControl with datatemplated ContentControl throws binding errors when its DataContext changes

In my project, I wanted a UserControl that can display different formats of an image (bitmap, svg), selected from a ListBox. The SelectedItem of the ListBox is bound to the appropriate view model, which in turn changes the DataContext of the UserControl, and what I want to achieve is for it to change the displaying control (an Image for bitmaps, a SharpVectors.SvgViewBox for svg files) through data templates. It does so, but raises data binding errors, as if the templates were still intact whilst the UserControl's DataContext has already been changed.

I should like to a) avoid any data binding errors even if they cause no visible problems b) understand what is happening, so I prepared a MWE, which, to my surprise, displays the same behaviour, so I can present it here.

My minimal UserControl is as follows:

<UserControl x:Class="BindingDataTemplateMWE.VersatileControl"
             xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
             xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
             xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006" 
             xmlns:d="http://schemas.microsoft.com/expression/blend/2008" 
             xmlns:system="clr-namespace:System;assembly=mscorlib"
             mc:Ignorable="d" 
             d:DesignHeight="450" d:DesignWidth="800">
    <UserControl.Resources>
        <DataTemplate DataType="{x:Type system:String}">
            <TextBlock Text="{Binding Content.Length, RelativeSource={RelativeSource Mode=FindAncestor, AncestorType=ContentControl}}" />
        </DataTemplate>
        <DataTemplate DataType="{x:Type system:DateTime}">
            <TextBlock Text="{Binding Content.DayOfWeek, RelativeSource={RelativeSource Mode=FindAncestor, AncestorType=ContentControl}}" />
        </DataTemplate>
    </UserControl.Resources>
    <Grid>
        <ContentControl
                Content="{Binding .}" />
    </Grid>
</UserControl>

The MainWindow that references this UserControl has the following XAML:

<Window x:Class="BindingDataTemplateMWE.MainWindow"
        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:BindingDataTemplateMWE"
        mc:Ignorable="d"
        Title="MainWindow" Height="450" Width="800">
    <Grid>
        <Grid.ColumnDefinitions>
            <ColumnDefinition Width="*" />
            <ColumnDefinition Width="*" />
        </Grid.ColumnDefinitions>
        <ListBox
            Grid.Column="0"
            SelectedItem="{Binding Selected}"
            ItemsSource="{Binding Items}" />
        <local:VersatileControl
            Grid.Column="1"
            DataContext="{Binding Selected}" />
    </Grid>
</Window>

with the following code-behind (to make the MWE indeed minimal, I made the window its own DataContext, but originally there is a dedicated view model):

using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Windows;

namespace BindingDataTemplateMWE
{
    public partial class MainWindow : Window, INotifyPropertyChanged
    {
        private object selected;

        public event PropertyChangedEventHandler PropertyChanged;

        public List<object> Items { get; }

        public object Selected {
            get { return selected; }
            set {
                selected = value;
                PropertyChanged?.Invoke(this, new PropertyChangedEventArgs(nameof(Selected)));
            }
        }

        public MainWindow()
        {
            InitializeComponent();
            DataContext = this;
            Items = new List<object>() { "a string", DateTime.Now, "another string" };
        }
    }
}

When I select a different item in the list, the desired effect takes place: the UserControl displays the length if a string is selected and the day of week when a DateTime. Still, I get the following binding error when selecting a DateTime after a string:

Length property not found on object of type DateTime.

and conversely, selecting a string after a DateTime yields

DayOfWeek property not found on object of type String.

It is clear that what I am doing is not meant to be done, but I do not know what the correct paradigm is and what happens in the background. Please advise me. Thank you.