Working Of a Tunnel Diode
According to this lecture in a reverse-biased tunnel diode the current is due to the alignment of conduction band in the p-region and the valence band in the n-region. Because of that, the electrons in the valence band of the p-region are tunneled to the holes in the valence band of the n-region.

Since this current has the same characteristic of the zener diode, it's considered as a zener with zero breakdown voltage. So the physical reason is the 2) but the alignment of the bands is due to the dopant concentration, as you say in 1).