Bright Man has an attack similar to Protectos the Goreroid from Mega Man ZX, where he launches a powerful explosive towards the wall. To avoid the resulting attack, you have to stand in Bright Man's shadow, similar to the way you had to avoid Protectos' nuke.
This attack also bears similarity to the attack used by Mr. Shine and Mr. Bright from Kirby's Dream Land 2.
Spark. Manbow is Kine's Spark ability from the same game.
Using Recycle Inhaler on the purple Up'N'Down enemies causes them to give chase to Mega Man for a while before exploding, in a similar fashion to the Scarfy enemy from the Kirby series.
Shadow Man uses Edge's Ninjutsu skills (Katon, Suijin, and Raijin) in Pharaoh Man's stage.
Hell Wheel is basically Kirby's Wheel ability. The very end of the game completes the reference.
The sprite for the Wire power-up on the weapon select screen is the Hookshot.
Square Machine uses an attack pattern very similar to Serges. It's even called the Serges Devil Machine. The Aqua Devil fight has some elements to the Agile fight in the same game (to a much lesser degree).
Nightmare Suzy is based on Nightmare Mother from Mega Man X6.
In its final phase, the boss (as a single eye) surrounds itself with four bubbles, making it resemble the Kirby boss Kracko Jr. Before that, it also drops blocks in the same formation as Tetris.
In the second half of Wily Stage 1, Mega Man can hijack a Sniper Armor, an Apache Joe Chopper, Gunner Joe spacecraft, and a Cannon Joe cannon. All of this is based on the Ride Chaser and Ride Armor sequences from Mega Man X.