The Greeks were very careful about drinking nearly always mixing their wine with water. Therefore is is surprise that they had undiluted wine at breakfast – or rather they dipped their bread in wine. My completely unsupported theory is that when making wine there was some that they rejected for drinking because it went vinegary rather than alcoholic and they used this for breakfast.
λέγει δὲ τὸ πρωινὸν ἔμβρωμα, ὃ ἡμεῖς ἀκρατισμὸν καλοῦμεν διὰ τὸ ἐν ἀκράτῳ βρέχειν καὶ προσίεσθαι ψωμούς…
He is referring to the early morning meal which we call breakfast (unmixed) because we soak bits of bread in wine unmixed with water and eat them.