The lists of reasons for milchigs is out of control. I heard a shiur once…mamish like 20-25 reasons, likely not an exhaustive list.
Nonetheless, the Rema in Shulchan Aruch gives the reason because one is supposed to eat both challahs in lechem mishna, as a rememberance the bringing of two loaves in the Beis Hamikdash. In other words, one will eat dairy, which now makes the bread unusable for a fleishig meal (din in Shulchan Aruch), then bentsch/wait/wash, eat fleishigs now with the other challah. Therefore milchigs is really at least outwardly just to remind one to eat both loaves of bread.
Again there are other reasons, but the most “codified” is worthy to take note.
DQB