A Few Similarities Between Megaman X4 and Megaman & Bass
First, Megaman X4 can be called "Megaman X and Zero" in this way. So Megaman X4 allows you to select a secondary character in Zero. If you play as either X or Zero, you will be forced to use the same character throughout the whole game. In Megaman & Bass, you can select either Megaman or Bass BUT you must play as the selected character throughout the whole game. Zero as we know it is also Dr. Wily's final creation which Dr. Wily's influence lived on through Zero in a way until Sigma took over what should have been Zero's position. Not to mention later in X7, Axl is introduced as a Bass alternative... though Dynamo can be considered a Bass tribute as well.
Gameplay-wise, Zero will have an easier time maneuvering stages once he's defeated Split Mushroom, now he can double jump (Bass can also double jump) while X can't. But the problem is while Zero is faster and may have an easier time maneuvering stages, he is limited to his new gameplay. Zero's primary weapon is a saber which while it is more powerful than X's buster, makes it hard to defeat bosses because you must fight in close range. Megaman X must focus on grabbing power-ups while Zero doesn't need them.
If Dr. Wily creates the ruse through his new robot King which he created to test Bass, Sigma also creates his own ruse by manipulating General of the Repliforce into leading a rebellion. Both King and General were manipulated by the villains, except Sigma's take it to a whole new level of moral event horizon. In the end, both King and General both wanted to atone for their crimes in the series. King in Megaman's ending shows he wants to atone for his sins. In Megaman X4, General actually sacrifices himself to stop the Final Weapon from destroying humanity.
For the bosses I would probably say these. Frost Walrus could be Coldman's counterpart, Burnerman would be Magma Dragoon's counterpart, Jet Stingray would be Pirateman's counterpart, Cyber Peacock would be Astroman's counterpart, Storm Owl would be Tenguman's counterpart (though Iw ould believe Storm Eagle is a Tenguman tribute), Slash Beast would be Groundman's counterpart, Split Mushroom would be Dynamoman's or Magicman's counterpart and Web Spider well... forced as it is would be Magicman's counterpart or Dynamoman's counterpart.