During the Space Dojo level, you cross through a pair of rather familiar shutters (which I forgot to mention when I first posted this) and encounter the character seen above as a boss called "Mecha Biker," a "Super Combat Bot" who comes in on a red jet-powered motorcycle. Once you knock him off, however, he takes to a more ground-based approach: Triple-shots, charge-shots, and even a sliding kick. All while this music plays:
And once you finish him, he even explodes in Mega Man's signature fashion.
All in all, it seems pretty obvious what they were going for here. Kudos to the folks at WayForward and Majesco for including this nod to the Blue Bomber!
And if you want to check the game out for yourself, Double Dragon Neon is available on Xbox Live Arcade and the PlayStation Network for 800 MS Points/$9.99, or for free if you're a PlayStation + member! Just be sure to bring a friend; the game is so much better in "bro-op."
Edit: In addition to a few other touches I forgot to mention (thanks to you guys in the comments for reminding me), GenjiRed even found a video of the level in question, one which is more platform-oriented than other parts of the game, so you can decide for yourself:
If you'd rather skip to the fight itself, click here or just skip to 8:50 on the player above.