I don't know where you got your numbers from, but the energy density this supercap has is on par with batteries: 20Wh/kg. Now look at the size of the caps she has. Those are samples that weigh grams. Whatever fits in a cellphone will weigh probably on the order of 100g, and will store on the order of 1 Wh. I don't see kilowatts for charging, never mind that you absolutely don't have to charge in 20s. A one minute charge cuts the power by a factor of 5, and anyone sane will go with what's economical and makes business sense, not with what was put on a headline somewhere.
Never mind that it doesn't matter much what the energy rate (power) is, what matters is the stored energy. 1 Watt-hour is 0.86 kg*Kelvin for water. Assuming whatever the phone and battery are made out of have the heat capacity comparable to water, releasing 1 W-h will heat things up by 10 degrees C.
In other words: yes, it's as harmless as any current battery technology. Get over it.
My other gripe: couldn't that girl publish this stuff like everyone else out there? All we have is stupid news articles and a single-page PDF summary. Weren't they supposed to have write-ups submitted with the project? WTF is all this stuff? To me it's not $5k-worthy science if it's secret.