I think there are good answers here: basically no one would claim to 'possess' or 'own' a coffee, which is exactly how it sounds in French. Context is key. I can't think of a direct analogy in English, but, for example, if I said "I own a dairy bar", you probably would not assume I was talking about a small ice cream treat. Or, if I said "My mother owns a diner", you probably would not assume that she has a human being locked in her basement that she likes to feed.
I put my "mother has a coffee" and it was accepted. I have doubts though if it should since it's quite a different meaning and to have a coffee in England most of the time means to drink a coffee. Someone can also have a coffee and not necessarily be drinking it. Can coffee be correct? Can it mean to drink and/or possess a cup of coffee?
You can possess a coffee without drinking it. In English you can 'have' a coffee and/or enjoy it, but in French, << tu prends/vous prenez >> (you take) <<un café et puis tu boit/vous buvez le café >>, so either she's not drinking and simply is in possession of it or (more likely) she owns a shop where they sell it. That's my impression, anyways.
I guess it's just not something that people tend to say. You'd say "my mother has a coffee" which should be accepted as a correct answer. However here, it uses "possède" which specifically means "to own [something]", so in this case you'd translate it as "my mother owns a café".
This word possède has come up in maybe one other exercise for me. This is the second time I get it wrong I had already forgotten which translation was required. You can't make the argument that you can't own a coffee because the translation given is my mother has a café. I could have also said my mother possesses a coffee or My mother possesses a cafe. Both make sense. This is a very poor sample sentence, especially since what we're actually testing for in this section is the proper translation of the word for mother.