The way the sentence is said, emphasis is put on the word ένα, so it means something like "We are reading one book" (not two, not five, one). In order for ένα (ένας, μία, etc.) to mean "a/an", emphasis should be put on the noun, βιβλίο in this case. Then it would mean "We are reading a book" (not a letter, not a newspaper, but a book).
I appreciate that, not that the audio was bad :) but that you reported it. There's a better chance of it's being corrected, once correction time comes around. It will but I remember with the Italian it was a nightmare then one day we couldn't believe our ears. We'd almost gotten used to the other.
I think you're confused about your grammar terminology.
A possessive is a word such as "my, his, our".
A noun is a "person, place, or thing", e.g. "book, democracy".
διαβάζω (read) is a verb -- an action word.
Words such as "I, you, we, they" are personal pronouns.
The ending on a verb changes to show who the subject is -- whether it's "I" or "we" etc.:
- (εγώ) διαβάζ·ω "I am reading"
- (εσύ) διαβάζ·εις "You [one person] are reading"
- (αυτός, αυτή, αυτό) διαβάζ·ει "He/she/it is reading"
- (εμείς) διαβάζ·ουμε "We are reading"
- (εσείς) διαβάζ·ετε "You [several people] are reading"
- (αυτοί, αυτές, αυτά) διαβάζ·ουν "They are reading"
These endings ·ω ·εις ·ει ·ουμε ·ετε ·ουν are regular for a large group of Greek verbs (but not all of them).
I put the subject pronouns in parentheses because they can often be omitted -- the verb ending will often be sufficient to show whether the subject is "I" or "we" etc.
In the previous question, I've heard the speaker pronounce the Greek word for rice ('ριζι') as /rihi/. Is this correct or is there something wrong with the audio?. Plus, the usual discussion button wasn't available for that question, and that's why I am posting this question here now.
"ρύζι" is pronounced quite simply "reezee" if the previous sentence sounded differently it was incorrect. This is a computer generated audio so some words are not exact. Fortunately, Greek pronunciation is very straight forward. What you see is what you get. Take a look at the Tips & Notes from the ABC skill for some ideas on the pronunciation of each letter. There are a few diphthongs and double consonants but not too many.
Duo has been experiencing some internet issues lately most have been corrected but perhaps the comments are not all in place yet. It was a good idea to come here.