Snapping to tangency points only works as intended when drawing a line from a point to a circle.

The tangency points QCAD (and other CAD systems) snap to are defined as the tangent of the line from the last defined point (= position of relative zero point, for example the start point of a line) to the circle, arc or ellipse.

Snapping to tangency points only works as intended when drawing a line from a point to a circle.

Ok!

When I try to make a tangential snap of an existing line to a circle by moving one line end to the circle, qcad snaps to a wrong snap point on the circle.

Still confusing to me is that if I try this with the Auto Snap function the Info Text on the mouse pointer will wrongly tell me that the "Tangents point" is correctly located ... ;-)

Well, it is a tangency point, just not the one you think (see red line from relative zero point to circle).
It might help to know that the object snap tools are unaware of what you are currently drawing or if you are moving something, pasting something, etc. The tangent and perpendicular snap always simply relate to the relative zero point (typically the last defined point).