Licking an anemone sounds like an unlikely scenario, but assuming that it was done, it sounds highly plausible that it could cause the kind of reaction you describe.
Some allergic reactions can cause a child's throat to swell enough to cause problems breathing.
Certain houseplants pose the same risk to young children. They chew on a leaf, their throat swells up, and they have trouble breathing.
BTW, you said you heard this from your marine biology teacher, right? Why the heck would you discount her statement, and think you'd get better information here? :huh: