if you look at your output window you should see the binding exception.
The problem you have is the following: within your usercontrol you will bind the label to the DP ProtocolNumber of your usercontrol and not the DataContext
, so you have to add for example the element name to the binding.
<UserControl Name="MainOptionsPanel"
x:Class="ExperienceMainControls.MainControls"
xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
x:Name="uc"
>
<Label Height="Auto" Name="numberLabel">Protocol:</Label>
<Label Content="{Binding Path=ProtocolNumber, ElementName=uc}" Name="protocolNumberLabel"/>
(...)
</UserControl>
EDIT: to clear some things up, your usercontrol also works if you change the binding in your MainWindow. but you have to bind to the DataContext of the MainWindow with RelativeSource.
<expControl:MainControls ProtocolNumber="{Binding Path=Number, RelativeSource={RelativeSource AncestorType={x:Type Window}}}" />