Is there any way to for the ToolTip to show up only on the TabItem header, but nowhere else?
You should only apply Tooltip
to the Header
not the whole TabItem
so change it to:
<TabItem>
<TabItem.Header>
<TextBlock Text="Tab1"
ToolTip="Tooltip of tab1"/>
</TabItem.Header>
<StackPanel>
<TextBlock Text="Content of tab1 with its own tooltip"
ToolTip="Tooltip on content of tab1"/>
<TextBlock Text="more content of tab1" />
</StackPanel>
</TabItem>