"We have been waiting," should not be accepted. "have been waiting" is the present perfect, and the perfect aspect means that it indeed has been completed: e.g. "Where were you? We have been waiting for five weeks, but you never showed up!" To translate the present perfect tense to Italian, you would use the passato prossimo «abbiamo aspetatto» or «siamo stato aspettando».
On the other hand, the present progressive tense indicates that the action has not been completed; that would be "We are waiting" in English and «Stiamo aspettando» in Italian.
I think that the other sentence in this section which starts with "Aspettiamo" is confusing some people.
The other lesson asks us to translate "Aspettiamo da circa sessant'anni" and only accepts the present perfect tense, for example: "We've been waiting for about sixty years".
So a Duolingo user can get that question first and then think that they need to say "We've been waiting five weeks" for this exercise, which isn't accepted.
The difference between the two lessons is all in the "da".
"Da" usually translates to "from", but when used in this type of sentence it is more typical to say "for" in English. For example, "Lavoro da due ore" would become "I've been working for two hours".
So, the use of the "da+a time period" in the Italian sentence will often shift the English translation from the present tense to the present perfect tense.
It can be confusing, but when you see the "Da + time period" remember it means that the action in the sentence has been going on for that time period.
The current exercise, doesn't have the "da", so it is just a present tense sentence. "We're waiting/We wait five weeks."
I have found the thread where this is the case.
This sentence is an idiomatic use of the present tense. It shows that, in such a sentence where you have waited for a while, you can use the present tense in Italian; in English, you cannot say "we wait for sixty years" without it sounding habitual. The Italian sentence does not necessarily imply that "we" habitually wait for sixty years.
It is the same as when you have the sentence «Ho diciotto anni.», and you would not translate this to "I have eighteen years."
Other than in idiomatic use, «aspettiamo» is translated as "we wait" or "we are waiting." Context is key.
The only way the English translation "We wait five weeks" would be correct is if this is something we do regularly, e.g. "Every time there is a new lottery, we wait five weeks, and then we buy a ticket." Without that indication of repeated action, this needs to be either present perfect progressive tense (We have been waiting) or future (We will wait). I don't know enough Italian, however, to know which of those is correct.
This Italian course makes this error quite frequently in that it translates something which is present tense in Italian into the present tense in English without realizing that the present tense in English means either something that is a general fact that is always true or some kind of repeated or regular action.