Yes, because the omega in the end of πίνω shows that it's referring to the first person (in singular). Τhat is, the person who's saying it.
EDIT: A few examples with other verbs too. Possible answers to the question «Τι κάνεις;» (What's up?), besides the common «Καλά είμαι» (I'm fine), can be: «Πίνω μπύρα» (I'm drinking beer), «Τρώω» (I'm eating), «Μινάρω/Παπαρίζομαι» (I'm killing time), «Παίζω WoW» (I'm playing W.o.W.), «Διαβάζω/Μελετάω» (I'm studying for school/university). As you can see the personal pronoun is being omitted in all these cases and it's normal.
EDIT2: «Μελετάω» isn't used that much in common speech even if it's closer to "I'm studying" than «διαβάζω». The latter has a literal translation as "I'm reading". Examples: «Διαβάζω μια εφημερίδα» (I'm reading a newspaper). You'll rarely hear someone use the verb «μελετάω» with a soft read as a newspaper or a magazine.
Just to add to what miliarma said, Greek is what's technically called a 'pro-drop' language, meaning that the verb endings are different enough that the pronouns aren't needed and can be dropped most of the time without confusing anyone. Some languages like Spanish and Italian are the same.Others, such as English, German and French always need the pronoun because the verbs are too similar sounding - ie if we said "drank water today", it could mean any person did it (I, you, he/she/it, we, they). We need one of those pronouns to clarify which we meant.
You're welcome! If you also wonder what happened to άρτος (bread in ancient Greek), ψωμί comes from the word ψώμος which meant a mouthful of bread or food in ancient Greek :) Although basic words have changed over time, their etymology is somehow related to the original word.
Just checked and Wiktionary says they both come from the PIE root *peH3- "to drink", just with different root extensions. https://en.wiktionary.org/wiki/Reconstruction:Proto-Indo-European/peh%E2%82%83-
The accusative is always the same as the nominative for neuter nouns, so νερό is the nominative or the accusative case.
νερά would be nominative or accusative plural, i.e. "waters" -- not often used, since water is usually treated as an uncountable noun.
And it's πίνω for "I drink", not πίρο.