Mugen is owned and run by Hirotoshi Honda, the eldest son of Soichiro Honda, the man who founded the Honda Motor Company. It has operated since the early 1970s preparing competition engines for Honda, often with unofficial help from the main Honda firm.
So that dude was partially right. Mugen was started by the same people who started Honda. I wouldn't go quite as far as to say its like "STi > Subaru", "SVT > Ford", etc, but Honda and Mugen are very closely intertwined.