How are there two C3 rotation axes in ammonia?
In a character table for $C_{3v}$, $ 2C_3$, means that the element is actually 2 $C_3$ elements in the same conjugacy class.  That means the two $C_3$ operations are related in some way.
In fact, it means they share the same single axis.  One $C_3$ operation is a $120^\circ$ turn around the axis, and the other $C_3$ operation is a $240^\circ$ turn around the same axis in the same direction - or equivalently a $120^\circ$ turn around the same axis in the opposite direction.
In the image below the two elements $C_3$ and $C_3^{-1}$ are the two symmetry elements of the $2C_3$ class, which are rotations about the same axis out of the plain of the image.

Symmetry elements of the $C_{3v}$ point group
Image taken from Fig S5 in supplementary information of T. Karin, et al, Phys. Rev. B 94, 041201