I'm no Mugen expert, but Hirotoshi's father founded Honda Motor Co, and Hirotoshi wanted his own business, so the Mugen-Honda relationship is a family one. As for ownership, I'm sure a shitload of Honda money is in Mugen, how much though, I'm not sure.
Honda cancelled it's funding for it's F1 engines in 1992, and Mugen picked up the F1 left-overs and started the Mugen-Honda F1 engine supplier.