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 Lunar Eclipse attack used by Mr. Shine and Mr. Bright from Kirby's Dream Land 2, where you had to stand in Mr. Bright's shadow to avoid getting fried.
Spark. Manbow is Kine's Spark ability from the same game.
If the name "Spark Manbow" sounds odd for a weapon that basically consists of firing light bulbs, it's because it's a reference to the Mega Man X boss, Spark Mandrill.
"Manbow" is also pronounced similarly to the Japanese name of the ocean sunfish, the same species as Kine.
Using Recycle Inhaler on the Up'N'Down enemies in Pharaoh Man's stage 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.
Upon defeating an end of stage boss, a victory ditty from the Pokémon series is played.
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.
To complete the reference, Mega Man uses Hell Wheel (a decidedly Wheel-esque power) for this section.
The section after Wily Capsule Boss Fight is based the organic final level of Gradius 2.
The Wily Buster is basically Samus's Hyper Beam from Super Metroid.
The credits have the theme of Fire Emblem. Like in that game's credits, it counts the combined amount of times the unit (or in this case, boss), won and lost a fight.
When you do a critical hit with any weapon, the damage is triple the usual amount. In the Fire Emblem series, any attack with a critical hit gets a multiplier of 3.
One type of Up'N'Down, according to the credits, is named Dig'N'Dugn.
The Mechanical Arena is basically The Arena from Kirby Super Star, with four health pickups in the hub area.
It tends to be more like "Helper to Hero" from the DS remake as the bosses are fought in a pre-determined order (The Robot Masters are fought in order of their DWN), and mini-bosses aren't fought.
As of the 0.97 release, it has the TRUE Arena, which likewise based on the remake of Kirby Super Star. This arena does have randomized matches.
One boss in the True Mechanical Arena is Green Helmet, a green Met not seen anywhere else. It has more health than regular Mets, but can be defeated with just one use of the Recycle Inhale, just like the Arena Waddle Dee from Kirby Super Star.