Most EFL/ESL teachers cannot explain article usage, so they often make up reasons on the spot, or they give out oversimplified "rules". This teacher might have just thought the sentence sounded better with a. However,
My country has three seasons - winter, summer, and rainy season.
is fine as it is. I would probably not use a here, since it is indefinite but we're talking about a definite season that comes each year. We're not talking about one (a) rainy season out of many possible rainy seasons. That's the most important part.
In addition, the phrase rainy season without an article acts as sort of a name, just like winter and summer, and we wouldn't put a before winter or summer here. So it would be something like
There are three people in class: John, Mary, and tall person.
If we want to talk about tall person as one among many, we would use a tall person. In the less likely case that tall person is actually the nickname of the third person, we wouldn't use a. This is sort of how rainy season works in your sentence, as the name of the third season.
It's also an extreme oversimplification that we use the with superlatives. We can, but a is also possible:
That was a most wonderful cake, Sammy, thanks for cooking it.
In general, ESL/EFL teachers have to give out some sort of rules as s starting point to learn article usage, but unfortunately these rules are usually way oversimplified and rarely, if ever, cover all uses of articles. And so articles remain troublesome for many learners for years and years. But one thing to do is not trust any list of simple rules about article usage. No one has ever come up with an explanation that describes all usages of the articles in English.