“Le” is definite article, so when your are saying “Elle mange LE petit-déjeuner”, you are making reference to some precise breakfast, known by both you and your interlocutor. That’s why without any context, “Le petit-déjeuner” refers to the idea of breakfast, in general, and this is what your teacher meant. I think the corresponding sentence in English is “She’s having the breakfast”, that is odd too (and might be incorrect).

What we want, to traduce “She’s having breakfast”, is to use the indefinite article (equivalent to english “a”, undertoned in this english sentence) : “Elle mange UN petit-déjeuner”

“Elle mange SON petit-déjeuner”, also used to translate “She’s having breakfast”, is more usual than “Elle mange UN petit-déjeuner”, without different meaning.

To conclude, “Le petit-déjeuner” could also refer to the food of the breakfast. So in some context, for example, “Elle mange le petit-déjeuner que son père lui a préparé”, the use of definite article “le” is used to specify what food she is eating.

It is true that French people don’t say “Elle mange le petit-dejeuner”. But the reason is not that this would mean “the idea of breakfast”. The sentence is perfectly comprehensible, it’s just not idiomatic at all. We might say “elle mange le petit déjeuner que j’ai préparé pour son frère” (she’s eating the breakfast I prepared for her brother).

The idiomatic way to say that someone is eating is to use the verb.

Elle déjeune. Elle est en train de déjeuner.   She’s having lunch.
Elle dîne. Elle est en train de dîner.   She’s having dinner.

The phrasing with “en train de” emphasises that the action is currently ongoing. The straight use of the verb can mean either that she’s currently having the meal or that it’s a habitual action, i.e. it can correspond either to “she has …” or “she’s having …”.

For breakfast, there’s a difficulty: the verb is déjeuner, which is identical with lunch. If there’s no risk of confusion, we’d just say “elle déjeune”. If there’s a risk of confusion, we might say “*prendre son petit déjeuner”, e.g.

Il est onze heure mais elle est encore en train de prendre son petit déjeuner.   It’s eleven o’clock but she’s still having breakfast.

Sometimes petit-déjeuner is used as a verb, which resolves the ambiguity, but it isn’t very common and may be considered non-standard.