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.
To further cement in the reference, Toad Man's stage theme is the overworld theme from that game.
When Mega Man gets the Toad Spell, the Weapon Get Screen is a green orb on the left of the word Toad, just like how Green Magic is defined in some earlier Final Fantasy games.
When Mega Man uses Toad Spell, the stars from the casting animation from the Famicom version of Final Fantasy III surround him. And the fact that the toads are easily killed with contact is a reference to the fact that Toad status effect lowers defense in the Final Fantasy series. It also references the ability of the Toad Spell to instant-kill foes in Final Fantasy II
The Wish Star Adapter, Rush Search and Rush Cannon are homages to Rinoa's limit breaks.
The Trio of Ring-Rings in Ring Man's stage have the same AI patterns as Lololo and Lalala.
The way to unlock the secret path to Wave Man in Dive Man's stage looks a lot like Samus using the Power Bomb to unlock Maridia.
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 (Flame, Blitz, and Flood) 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, the Evolution Complete theme 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 Aqua Devil (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 II for the Famicom.
The Wily Buster is basically Samus's Hyper Beam from Super Metroid.
The Item Collect theme from Super Metroid is also used when the player acquires the Balloon and Wire Adaptors.
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 Kirby Super Star Ultra as the bosses are fought in a pre-determined order (The Robot Masters are fought in order of their DWN/DCN), and mini-bosses aren't fought. In a strange twist, Kirby: Planet Robobot has an Arena... that's mechanically themed (just like the rest of the game) and the bosses are fought in order. You also fight mini-bosses, but the Arena from that game is literally a Mechanical Arena, along with having aspects from the Arena from this game.
As of the 0.97 release, it has the TRUE Arena, which is likewise based on Kirby Super Star Ultra. This arena does have randomized matches and includes all the minibuses as well.
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 Inhaler, just like the Arena Waddle Dee from Kirby Super Star.