My WPF control cannot read app.config

Step 1: Make a WPF Class Library project to contain the class you want to create.
Here I create a new control, that inherits from Button, but sets its content from a config value as you want (exception handling etc. not shown):

(Make sure your project references System.Configuration)

using System.Configuration;
using System.Windows.Controls;

namespace AppConfigControl
{
    public class AppConfigButtonControl : Button
    {
        public AppConfigButtonControl() : base()
        {
            var configValue = ConfigurationManager.AppSettings["ButtonContent"];
            if (null != configValue)
            {            // Example: Display the setting in a TextBlock
                var textBlock = new TextBlock { Text = configValue };
                this.Content = configValue;
            }
        }
    }
}

Make a new (consumer) wpf project that either refrences the AppConfigControl control or its compiled dll. Make a view that contains your new control:

Window x:Class="AppConfigControlConsumer.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:AppConfigControlConsumer"
        xmlns:controls="clr-namespace:AppConfigControl;assembly=AppConfigControl"
        mc:Ignorable="d"
        Title="MainWindow" Height="450" Width="800">
    <Grid>
        <controls:AppConfigButtonControl Margin="20" Height="40" Width ="150"/>
    </Grid>
</Window>

- Notice the line: xmlns:controls="clr-namespace:AppConfigControl;assembly=AppConfigControl"

Make a new App.xaml file for your configuration:

<?xml version="1.0" encoding="utf-8" ?>
<configuration>
	<appSettings>
	<add key="ButtonContent" value="Config text value"/>
	</appSettings>
</configuration>

Run and enjoy