How to bind local property on control in WPF

Try:

Content="{Binding ElementName=_this, Path=CompanyName}"

Without the DataContext binding.

EDIT

I have no problems with your code, have named your window to x:Name="_this"?

<Window x:Class="WpfStackOverflowSpielWiese.Window3"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        Title="Window3"
        Height="300"
        Width="300"
        x:Name="_this">
  <Grid>
    <StackPanel>
      <Button HorizontalAlignment="Center"
              Name="btnChange"
              Click="btnChange_Click"
              Content="Click Me" />

      <Label Name="lblCompanyId"
             HorizontalAlignment="Center"
             DataContext="{Binding ElementName=_this}"
             Content="{Binding Path=CompanyName}"></Label>

    </StackPanel>
  </Grid>
</Window>

Is your window really Window3?

public partial class Window3 : Window
{
  public Window3() {
    this.InitializeComponent();
  }

  public static readonly DependencyProperty CompanyNameProperty =
    DependencyProperty.Register("CompanyName", typeof(string), typeof(Window3), new UIPropertyMetadata(string.Empty));

  public string CompanyName {
    get { return (string)this.GetValue(CompanyNameProperty); }
    set { this.SetValue(CompanyNameProperty, value); }
  }

  private void btnChange_Click(object sender, RoutedEventArgs e) {
    this.CompanyName = "This is new company from code behind";
  }
}