The difference is between English as used in daily practice and English used only following the rules of grammar.
Soolrak is correct that "coffee" is an uncountable noun, and, when following the strictest rules of grammar, should never have an indefinite article in front of it.
With that said ...
Linguistic shortcuts are taken every day in every language in every country. (The only stagnant languages are dead languages.) Asking for an uncountable liquid with an indefinite article in front is one of the single-most common ones you will find all across the United States and, according to many British people, in the UK.
Context is critical here. We are in a segment on restaurants. Locations and situations often have a dialogue in English unique to the environment. I have worked in restaurants in 4 different states and traveled in all but 3. If a server asks a table "What would you like to drink?", they will most often be answered with "a coffee", "an orange juice", "a sweet tea", "a Coke", "two lattes to go", etc.
What we mean is "a cup of coffee", "a glass of orange juice", "a glass of sweet tea", etc. However, excepting wine, they will most likely be answered in the shorter form.
FULL DISCLOSURE: Native English speaker - US, Southern Appalachian dialect. Other uses of English may vary. Advice about Spanish should be taken with a grain of salt.
Well I'm am a native English speaker and if I was ordering two coffees, I'd ask for two coffees!
The only reason you're not making coffee plural is because you're making cups plural.
Two coffees please.
Two cups of coffee please.
One cappucino, one small americano and two espressos please.
Why do you have to ask for it in a cup?
Do you really go into Starbucks and ask for a cup of coffee please?
Yes, in my dialect (US Pacific Northwest), it would be perfectly normal to say, "I drank three coffees today." I do think it carries a slight connotation of having purchased the coffees from a coffee shop, rather than made them yourself, but it could definitely be used either way.
Cream and milk are two completely different things. In US "coffee with cream" and "coffee with milk" are both perfectly acceptable. Depending where you're located in the US, your coffee may already come with milk if you don't specifically ask for it without, as in 'black coffee'.
I know that technically this is a milky coffee but no one I know ever asks for milk when they order a coffee. They just order a coffee and it's assumed you want milk unless you order a black coffee so realistically Duo should accept simply "a coffee" as an answer seeing as native speakers of English will instinctively understand that. I mean whenever I've learnt any form of Romance language we have always been taught that we must specifically add the equivalent of "with milk" because otherwise we would be given a black coffee instead, showing that it is clearly commonplace, at least in the UK, to ask for "a coffee" and expect it to come with milk
I don't drink coffee myself, but I've been around a really, really long time. I've lived in a lot of places - Ohio, Iowa, UK, Texas. I've traveled around the world. And I've NEVER heard anyone ask for milk with their coffee. Not in English. En español, si. En français, bien sûr. But not in English. They always ask for "coffee, white" or "coffee with cream".
I don't think I have ever heard anyone say " an orange juice please" or "a coffee with milk" - I have always been told to translate the way we speak. We just ask for "orange juice, please OR a GLASS of orange juice" and we say "coffee with milk OR maybe a CUP of coffee with milk", but the indefinite articles of "a" or "an" probably never used without saying the object the liquid is being put into