So, according to this,
there are four different recent standards that allow color emoji in font files – I suspect that none existed when Processing font support was developed.
SVG-in-Opentype
COLR/CPAL
SBIX
CBDT/CBLC
If you could get the svg from opentype then you could get the color image – assuming the svg could be read by PShape. But you might still need to handle text size and kerning / leading.
Possible that emoji4j could help with that? Haven’t looked at it, but it seems like an external library that makes svg retrieval easy would be the first step. Or updating Processing core font components.