Plastic dishes are prone to damage and scratching around the rim of the bowl. Over time, these scratches get deeper and deeper and begin to collect bacteria, no matter how well they are washed. The bacteria proliferates and eventually can cause problems for your pets, i.e. feline acne on kitty chins, among other things. I have never heard of a direct correlation between eye and muzzle staining and plastic dishes, but it stands to reason that yeast bacteria could also collect and grow on the bowl, which would cause some of the problems to which you're referring.
Play it safe and stick with stainless steel. It's easy to wash, easy to decontaminate, and lasts damn near forever.
Guard well within yourself that treasure, kindness. Know how to give without hesitation, how to lose without regret, how to acquire without meanness.
George Sand (1804 - 1876)