Can household dish soaps, stored diluted, breed dangerous bacteria?
Molds can often live off even undiluted detergents.

I had discarded contaminated opened bottles more than once, one in recent weeks. There was growth near the nozzle and black streaks suspended in the liquid body. Given the viscosity of typical detergents, it is probable that droplets of contaminated water nurtured the growth and perhaps not the detergent itself. Either way, I would end up dispensing microbes, pathogens or not. Bigger concern is what can be lurking in other bottles without visible signs.

A quick search came up with this "Danger In The Soap Dispenser" article on albeit hand soaps rather than dishwashing soaps and bacteria rather than fungi which are sometimes more resilient.