In informal contexts the more common Germanic rooted word is certainly more heard. If I were talking to a friend I certainly would be more likely to say gotten, but if I were talking to a bank or a company which was supposed to send me money I would more likely say received. I do think that Duo's has a strange take on cognates in general. They use the cognates almost exclusively with words like recibir and obtener, but then tend to ignore enormous as a translation for enorme in favor of huge. I agree that haven't gotten should be accepted here, although I understand the idea that pushing a truly close cognate has the advantage that you understand all its possible meanings and uses. Beyond the false cognates like embarazada there are a lot of partial cognates like realizar which has one shared related meaning but not the most common one. To the extent you can understand a Spanish word from the English word in all its senses the more of an advantage you have.
I see you toned down your post. The issue here is that, although In America we use both got and gotten, we don't use them in the same way. The past participle of get in its sense of to receive is gotten. Have got is generally actually essentially present tense. It is sometimes used to mean must. So I have got to go means I must go but I have gotten to go means that I have been allowed/been able to go. Additionally it is commonly used in an essentially redundant way. I have got a new car (or more likely contracted to I've got a new car) . This is present tense and different from either the simple past I got a new car or the present perfect I have gotten a new car. You will see this redundant usage on Duo sometimes, although even people who might well say it this way wouldn't write it that way. When American English and British English have different meanings for something, Duo does keep to its standard of American English. But they have been working to be more inclusive when it's not in direct conflict.
Got is already the past tense of the verb get, at least it is in this meaning. In American English as mentioned above have got is a colloquial way of saying have, although not a particularly sophisticated one. Of course there is also the expression I have got to which is again just an emphatic way of saying I have to. Since these differences can be misleading to students this is an example of a good reason for sticking to a single standard for interpretation. That standard on Duo is American English.
I have never seen Duo leave off the not in English where it belongs. If you saw I have received my money for this statement it is clearly wrong. This is a fairly straightforward sentence which basically should only be translated as I have not received my.money or I haven't received my money. Duo has recently been accepting and presenting contractions in English and maybe in that process an error crept in.
Duo does like to limit responses as much as possible, but in this case there is another issue as well. In this day of charge cards, checks and debit cards, cash tends to mean the bills and coins type of money. You can speak of transferring money or paying money without actually touching or dealing with these. In Spanish, to talk about this sort of cash you say either dinero en efectivo or simply el efectivo. For this reason, making the distinction between cash and money is helpful for students of Spanish.
Duo likes you to use the cognate when it is a clear match in terms of meaning. Received is a more specific term than got, but I agree that it should be accepted. Report it. But in order to try to simplify Duo's overcrowded servers, if you take the easy, obvious answer on Duo and let your own sense of English provide you with other full or partial good translations
It is true that not everyone speaks American English. It is also true that American English has more speakers, both native and second language speakers than UK English, although the latter has perhaps a greater geographic spread. Finally it is also true that having a language standard is both common and productive. There are a lot of Spanish language programs which use UK English as a standard, although they tend to teach Spanish from Spain. But Duo has American English as a standard, although they have been making an effort to include some of the British English terms. But grammar construction is more complex an issue. Here the standard is more important.