Setting a Property value to another property value in XAML

I am new to XAML, I want to change the background of a button to the color of the BorderBrush when I hover above it

Here's the code:

<Style TargetType="Button">
            <Setter Property="Template">
                <Setter.Value>
                    <ControlTemplate TargetType="Button">
                        <Border Background="{TemplateBinding Background}"
                        BorderBrush="{TemplateBinding BorderBrush}"
                        BorderThickness="{TemplateBinding BorderThickness}"
                        CornerRadius="15">
                            <ContentPresenter HorizontalAlignment="Center"
                                      VerticalAlignment="Center"/>
                        </Border>
                    </ControlTemplate>
                </Setter.Value>
            </Setter>
            <Style.Triggers>
                <Trigger Property="IsMouseOver" Value="True">
                    <Setter Property="Background" 
                    Value="{Binding BorderBrush, RelativeSource={RelativeSource AncestorType={x:Type Button}}}"/>
                    <Setter Property="Cursor" Value="Hand"/>
                </Trigger>
            </Style.Triggers>
        </Style>

The value of the Cursor changes as it should, the Background however doesn't.

I also tried RelativeSource={RelativeSource {x:Type BUtton}} but it didn't work.

You have to use RelativeSource.Self:

<Trigger Property="IsMouseOver" Value="True">
    <Setter Property="Background" 
            Value="{Binding BorderBrush, RelativeSource={RelativeSource Self}}"/>
    <Setter Property="Cursor" Value="Hand"/>
</Trigger>

Be aware however that the Background Setter in the Style Trigger would be ignored if your directly assign the Button's Background like e.g.

<Button Background="Transparent" .../>

In that case move the Trigger to the ControlTemplate, with RelativeSource.TemplatedParent:

<Style TargetType="Button">
    <Setter Property="Template">
        <Setter.Value>
            <ControlTemplate TargetType="Button">
                <Border x:Name="border"
                        Background="{TemplateBinding Background}"
                        BorderBrush="{TemplateBinding BorderBrush}"
                        BorderThickness="{TemplateBinding BorderThickness}"
                        CornerRadius="15">
                    <ContentPresenter HorizontalAlignment="Center"
                                      VerticalAlignment="Center"/>
                </Border>
                <ControlTemplate.Triggers>
                    <Trigger Property="IsMouseOver" Value="True">
                        <Setter TargetName="border"
                                Property="Background" 
                                Value="{Binding BorderBrush, RelativeSource={RelativeSource TemplatedParent}}"/>
                        <Setter TargetName="border"
                                Property="Cursor"
                                Value="Hand"/>
                    </Trigger>
                </ControlTemplate.Triggers>
            </ControlTemplate>
        </Setter.Value>
    </Setter>
</Style>