The RepeatButton in your PlaylistTabView.xaml represents the style of the Button which you circled out, so you can set the BorderThickness to 0 in it. Note that there are two RepeatButtons in your xaml, so you need to set them both.
PlaylistTabView.xaml:
<RepeatButton
x:Name="ScrollBackButton"
VerticalAlignment="Stretch"
Background="Transparent"
BorderThickness="0"
Delay="50"
FontFamily="Segoe MDL2 Assets"
Interval="100"
Style="{StaticResource RepeatButtonRevealStyle}"
Visibility="{Binding ScrollableWidth, Converter={StaticResource GreaterThanToleranceVisibilityConverter}, RelativeSource={RelativeSource TemplatedParent}, FallbackValue=Collapsed, TargetNullValue=Collapsed}">
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</RepeatButton>