Conservation of angular momentum in a planetary system

Why is angular momentum conserved when a planet revolves about sun in an elliptical orbit? Why is linear momentum not conserved in this case?

$$\rm \text{no external }\color{red}{torque}\to\color{red}{angular}\text{ momentum conserved}\\ \text{no external }\color{red}{force}\to\color{red}{linear} \text{ momentum conserved}\\$$

There is no external torque about the sun since the force of the sun and position vector are always at an angle $180^\circ$ since $\bar \tau =\bar r\times\bar F$, so angular momentum is conserved.

But since the path in not circular but elliptical, the position vector is not perpendicular to direction of motion hence some work is done, which changes the momentum indirectly by changing magnitude of velocity directly.

Both are conserved if you consider the whole system: If the earth looses linear momentum, the sun will gain it and vice versa. Subsystems may violate conservation laws (e.g by transfering energy/momentum). This is called local violation. But globally conservation laws will always hold.

The question why they hold globally in the first place, can be answered by Noether's theorem if the laws of physics (that is the equations of motion) are form-invariant under a continious transformation.