There is a longstanding custom in Klal Yisrael to eat donuts on Chanukah, which dates back to the time of the Rambam’s father, Rabbi Maimon ben Yosef. Sefer Sarid Upalit (p. 8) cites his words:
"One should not be lenient with any custom, even a minor one... The tradition is to make fried dough, known as ‘Al Spalang’ in Arabic... This practice is ancient, as the dough is fried in oil to commemorate the blessing."
Rabbi Maimon is understood to mean that we eat donuts fried in oil to commemorate the miracle of the jug of oil.
In Halichos Shlomo (17:20), R’ Shlomo Zalman Auerbach offers another reason for the custom, based on the Gemara in Avoda Zarah (52b). After the victory, the Chashmonaim purified every part of the Beis HaMikdash, except for the Mizbeach. They were forced to bury the stones of the Mizbeach. Eating donuts gives us a unique opportunity to pray for the Mizbeach. This is because when we eat donuts, we recite Al HaMichya afterward.
In Al HaMichya, we ask Hashem for mercy on several things, including the Mizbeach, when we say “ve'al mizbechecha.” This phrase appears only in Al HaMichya and not in Birchas HaMazon. Therefore, the custom of eating donuts, which requires Al HaMichya, was developed to provide additional opportunities to pray for the Mizbeach.
A practical difference between these two reasons would arise in the case of baked donuts rather than fried ones. According to Rabbi Maimon’s reasoning, one would not fulfill the custom with baked donuts, as they are not fried in oil, which is the key element of his rationale.
However, according to R’ Shlomo Zalman Auerbach’s explanation, baked donuts would still fulfill the custom, as they are subject to the Al HaMichya blessing, which provides the opportunity to pray for the Mizbeach.