I think the numbers are close because the Milky Way and the Andromeda Galaxy make up almost the entire mass of the group (i.e. there are many other galaxies, but they are MUCH smaller).  The Karachentsev paper (where the local group estimate of 1.29×10^12 solar masses comes from) mentions 5.8×10^11 and 7.1×10^11 solar masses for the Milky Way and Andromeda, respectively.  Those estimates are somehow derived from the local group mass estimate, however, instead of a method focused solely on the Milky Way.