I don't understand why "purses" is incorrect . Purse/bag/handbag are all synonyms.
That's very interesting, because I did a google image search on "die Taschen" and 100% of the images were of purses. And I get that Handtasche is like handbag. (Not all bags are purses but all purses are bags). I come from a German speaking American family (back two generations) and this really confuses me. I'm sure you are right, but it is very puzzling to me. I also thought that Tasche meant pocket, but I don't see that reflected here. But the German I have been exposed to may be archaic, so perhaps that's the problem.
You are right, it also means pocket (and it is an accepted translation, not just shown as the first translation so you don't see it, but try it next time, if you will, you won't lose a heart ;) )
I think the difference here is that in English there is another word "purse", which does not kind of exist in German. The closest is "die Handtasche", which adds extra information to the "Tasche". All "Handtasche" are "Tasche", but not vice versa, it is a subcategory. Just like every "Esstisch" is a "Tisch", but not the other way around.
So I can call "Handtasche" "Tasche", just as I can call "Esstisch" "Tisch", but it does not mean that "Tasche" is a synonyme for "Handtasche". That is also why Google Image Search shows such photos, when searching for "Tasche" you will find all sort of "Taschen", all the "subcategories" too, which probably have their own words too.
Does that make sense?
I think the problem here is British English vs American English. As I understand it, in Britain a purse is a small bag for change, what we in the US call a change purse. For us, a purse is the same as a pocketbook or a handbag. Our usage is based on age and the area we are from.
Die Tasche = "the bag" (singular)
Die Taschen = "the bags" (plural)
If you didn't know this already, you do now :) This is often how Duolingo teaches. Otherwise, you can do some extra work yourself and look up the singular/plural forms of every new word that you learn.
Also, it's not a firm rule but more an occasionally useful hint... but feminine nouns often get an -(e)n ending in plural. So if you see die ...-en it might be a plural rather than singular feminine noun.
They're wonderful! But they have a separate word in German... Rucksack! (Plural: die Rucksäcke)
Interestingly, while in English you might describe a "rucksack/backpack" as 'a type of bag...', in German it is clearly a distinct concept! The official dictionary Duden does not describe it as a Tasche but as a "holder/container". Wikipedia is also useful for a discussion about what is a Tasche and what is a Rucksack.
Using an image search or German Wikipedia can often be very helpful at figuring out what is/isn't included in the meaning.
Singular: der Koffer. Plural: die Koffer.