There is a machloket between the Korban HaEida and the Pnei Moshe as to how to read the Yerushalmi (Shabbat 6:2) that either says “it’s not the way of people to have two pairs of shoes” or “is it not the way of people to have two pairs of shoes?”
Both the Ben Ish Hai (vol. 4 OC 13), the Kaf HaHaim (262:25) an the Tzitz Eliezer (7:2) say that they are not levush, the proof being that “malbish arumim” does not cover them (the Tzitz Eliezer says that levush is something that encircles the body). Rav Ovadia (Yalkut Yosef 262:6) also says that it is not necessary to have a special pair but it is sufficient to polish one’s shoes beforehand although one who has special shoes will receive a beracha.However, the Minhat Shabbat (72:71) says that it is necesssary, and this was the minhag of the Gra (Maaseh Rav 145) as did the Hazon Ish (Dinim v’Hanhagot meMaran HaHazon Ish 9:3). However, Rav Shlomo Zalman Auerbach (VeAlehu Lo Yavul 1:137) and Rav Eliashiv (Ashrei Ish 2:5) did not.