No-Coverage Answer
The Taygetan contact material does not discuss intergalactic governance or whether other galaxies have their own Federation equivalents. All discussion of the Galactic Federation pertains to our galaxy (the Milky Way), and the scope of Federation jurisdiction beyond our galaxy is never addressed.
The material does describe multiple levels of Federation — a local Federation around Earth, a broader galactic Federation, and a "Deep Federation" or "Galactic Quadrant level" Federation that occasionally audits local operations (S-114, S-161). But all of these operate within the Milky Way. The corpus does not discuss civilisations from other galaxies or intergalactic political structures.
See also: Where is the Galactic Federation based? and Is there a hierarchy of races within the Federation? (cover Federation structure within our galaxy)
Searched: All 834+ transcripts, enhanced topic index (grep for "galaxy admin," "intergalactic federation," "other galaxy," "different galaxy"). Zero substantive results.