Common English Answer
The effect on the Dalek says "Both." In common usage, and as it is defined, both means two items. Since you may discard both The Doctor and the Dalek card having the Effect, considering that the Dalek is one card, you may only cause 1 The Doctor to be discarded.
2 - 1 = 1
You only get to target one of The Doctor's in play with the Dalek effect.
The Player is required to have "The Doctor" and a "Dalek" to win with the Goal "EXTERMINATE!" The image for the Card shows only a Silhouette for The Doctor and a Dalek. The silhouette matches the same in the bottom left corner of each Doctor card. From this, any of the Doctors and a Delek would satisfy the Win condition.
Comparatively, the other Goals list a specific doctor.
Consider the FAQ, which covers the situation in which you have the option of multiple Doctors with which to discard with the Delek. If it was required to discard all Doctors with the Delek, the FAQ wouldn't provide you with the strategic option of targeting your opponent's Doctor.