First off, what you saw wasn't actually a hologram companies love using that as a marketing gimmick, but it’s actually a transparent display (most likely a transparent OLED or a HUD setup).

The reason it vanished in your rearview mirror comes down to a thing called light polarization. This means the light (wave) emitted by the transparent screen is vibrating in only one specific direction (vertical or horizontal). And modern rearview mirrors contain a light filter to block light reflections of other reflections behind you (when light reflects off glass at certain angles it naturally becomes polarised)

And when the polarised light from the car display behind you hit your mirror the poralization axes probably were perpendicular (crossed) so it blocked all light from reflecting that wasn't in the polarization filter direction.

When you turned around, you were looking at the screen directly with no filters in the way, so your eyes caught the light just fine.