"Man sieht sich" is a generally normal way of saying goodbye to someone, though not as common as "bis morgen/dann/spaeter/gleich". The benefit of it though, is that it sounds considerably more 'normal' than the others when specifying a location - "bis dann in der Schule" sounds a little unusual, somehow.
Because this is a different word. "Mann" is a noun, "der Mann" meaning" "the man".
"man" (note that it is not capitalized (except for at the beginning of a sentence)) is an indefinite pronoun that literally translates to "one" in English and very often is translated by "you".
And what would a German say when what he really means is:
One sees oneself in the school.
(reffering of course to the visual imagination). Or to make the sentence more lifelike:
After spending the whole day in the sweatshop, one just sees oneself sitting before the TV with a can of cold beer.
There isn't a "they" in this sentence, "man" functions as "one" (as in "one sees you"), or more generally as "you".
Besides that, idioms generally only have one answer, Duolingo is pretty rigid when it comes to them, it usually only wants the correct idiom translation rather than a literal one
This is an idiom.
"Man" is often used impersonally, like in the English sentences:
They say this is a good store. ("They" isn't actual people, it's just a general "they")
When you forget to water your flowers, they die. ("You" isn't actually referring to the addressee; it means "anyone".)