In keyboard settings (at least in Big Sur) this can also be rebound to tapping fn, which is quite useful for typing math.
For Windows users, 'WIN + .' does this also - however I've found some differences from the emoji graphic shown in the popup, to the graphic actually pasted in the application.
Some web-based applications (most notably Facebook and Twitter) will replace emoji characters with their own custom graphics to guarantee that what the poster sees is what the reader will see since emoji appearances can differ slightly between fonts and thus operating systems.