I wasn’t expecting this question to bring so many responses. Since it did bring some discussion, a few reflections from me. Not to deny the no-mold camp, just because I can’t stand somebody being wrong on the internet.
Heavy metals may be toxic to life, but the effect is localized to where they occur. I doubt they’re distributed across the entire sponge. I would expect to see them mostly in the middle and around the top.
Temperature kills mold, but it’s a localized effect.
The sponge is indeed moist, not being drowned in water. If the latter was the case, mold wouldn’t be a problem. Mold doesn’t thrive under water.(1)
I don’t know why is richard.cs’s friend keeping water in bottles, instead of using fresh water from tap, but I’m not sure if mixing fuel and hot solder tips is a good idea. On the other hand probably not a problem: IPA boiling point is lower than that of water, so it evaporates first, and also water is good at stopping fire.
(1) Except some species living inside animals. But all soldering sponges I’ve seen in my life seemed pretty dead.