What is that mould/fungus green stuff? Is it unsafe to my health?
It is definitely unsafe for anyone's health. Do not ingest it under any circumstances. Do not feed it to anyone else either.
If the surfaces of the mug / feeder are good, then normal washing should do it. Maybe more careful washing than usual.
For extra safety, you may let then sunk in a concentrated solution of baking soda. Also, have them cleaned with a good alcohol, or hydrogen peroxide.
Another simple disinfection method: just boil them for about 30 min. Pretty much everything will die. It will damage plastic, so be warned.
However, if they are ceramic (of any kind, or any porous material) and have cracks, fissures, dents, they might already have more-or-less dangerous colonies built inside the walls.
Things to try in this case, after proper cleaning (see above).
Alternative 1: first use you mug for 2-3 days. If nothings bad happens, use the feeder - maybe even on yourself first. Then on the child, monitoring the result. If nothing happens in 2-3 days, they are completely safe.
Alternative 2: especially if they are cheap, throw them away and buy new ones.
Note: I have no idea how the feeder looks like, I just assume it can be used by an adult.