Recently, I observed that there is some brown slimy stuff on the underside of my plates. I don't know what they are or how they form. I tried removing it with a tooth pick, but was too hard to come off. Does anyone know what this is? How can it be cleaned?
My first guess is oil deposits that have hardened through oxidation. Especially the highly unsaturated oils are prone to oxidation, which will after a while lead to insoluble deposits (like in the original oil paints, where the linseed oil polymerises after prolonged contact with air). It's hard to remove as it's not brittle, and deforms rather than breaking, like an oil varnish.
(are those items you show plates, or pan covers/lids?)
It doesn't look like any kind of mold.
The good news - that is on the under side of the plate, so even if toxic, it shouldn't touch your food. Bad news - that rolled edge holds moisture and grime and there's no way to clean inside of it.
Since you wash them so often, that rolled edge probably always has some moisture in it which can cause mildew to grow or cause the plate to corrode. Mildew is normally softer and should scrape away. If the deposits are very hard and resist scraping with something harder than a toothpick, it could be corrosion of the metal (rust, oxidation, etc).
Honestly, I would try to get plates that do not have a rolled lip like those. They will always be hard to clean and less than sanitary.