Actually there is an infinity of tangent vector at point P, but I get what you want. What you really want is the tangent of the center of your geometry (ie the red line on your schema), because the tangent of a line is a vector and not a plane. There is no way of computing this “center of geometry” line using existing ParaView filters AFAIK so as Mathieu said, you can just compute what you want manually using the python calculator.
Also note that although it’s easy to create multiple slices that have the same normal, it’s not trivial when they’re not the same, as it is in your case. You’d have to create a slice filter for each slice with a different normal.
In the end: there’s no existing filters or smart workaround for your use case AFAIK. You can always try to hack your way around by using the python calculator or by creating your own plugin.