I’m not 100% on this, but I think “consommé” is the French spelling, “consome” is the Mexican one.
In French cookery a consommé is very specifically a clarified broth that goes through several processes and ends up looking like tea. I believe in Mexican cookery, the consome that accompanies birria goes through far fewer processes and so retains more solids and therefore has a different texture/flavour and is more of a gravy.