“Dual mono” indicates two completely separate, independent, unrelated channels, usually in one device. For example, in a dual mono preamp, you might run a vocal through one channel with its own control settings and a bass guitar through the other channel with a completely different set of settings. The two signals would be routed to different places.
On the other hand, “stereo” indicates that the two channels are related and connected. For example, a stereo keyboard string patch contains two channels of related material that are intended to function as one stereo signal; you’d typically process and route those the two channels that comprise the stereo signal together.