(Link to the song https://www.youtube.com/watch?v=_qjnTfCJqqw).
And then it clicked, the boss of level 3 in Contra III is a giant Snatcher!

He even shoots beams from out of his mouth! Crazy coincidence isn't it!
(It should be noted that Contra 3 takes place in the year 2636 so there probably isn't any story relation to it. Also Contra III and the PC-Engine CD release of Snatcher came out in 1992.)