I'm a map enthusiast and I'm curious about the projection used in this map. Can you identify it?

This map appears to be using a Gnomonic projection (also known as Azimuthal Gnomonic). This is inferred from the fact that it shows the entire globe (or a significant portion of it, including the poles) and straight lines of longitude and latitude appear as curves, which is characteristic of projections that project the globe onto a tangent plane.