Why do most Ashkenazim who wash for bread for the Melave Malka say Shir Hamaalot and Migdol (instead of Magdil) during Bentching at a Melave Malka Motzei Shabbos?
Regarding Shir Hama'alos, there are those who say Al Naharos Bavel like any other weekday (Rav Chaim Kanievsky in Sefer Sh’eilas Rav page 338) or others who say Shir Hama’alos, as on Shabbos, as long as the benching was recited before Chatzos.
Regarding migdol vs. magdil, some authorities hold that one should say say "migdol yishuot malko" like on shabbat itself, instead of "magdil yishuot"[Kaf Hachaim 189:11], while others say that one should say the regular "magdil yishuot" on Melava Malka just like during the week. [Yalkut Yosef (Kitzur Shulchan Aruch 300:4]
Perhaps the reason for the more shabbos-like (shir hama'alos & migdol) representation in benching is due to the shabbos-like quality of the meal. Meaning, Rav Berel Wein has a well written piece describing "a meal of departure in honor of the Shabbos queen prepared and eaten in order to allow us to show our love and respect for this most royal of days on the Jewish calendar." Essentially, melava malka is the final extension of the shabbos meals before fully entering the rest of the week.