Actually Marco, in both cases here it means: time/ period. Chronic means: مزمن, like in chronic illness.
"ومنذ ذلك الوقت انحفر في ذاكرتي شيء اسمه الزمن: And since then, something called time was engraved in my memory."
I would say that the author was probably talking about how his life is becoming hard, and by time here, he meant his life in general. He's probably saying that his life's difficulties were engraved in his memory, yet it really depends on the whole context you are reading.
"ومنذ ذلك الوقت عرفت أي جنون هو الزمن، تحوّل الأكل في حلوقنا إلى نخالة: And since then I knew which insanity is time, food turned out to bran in our throats."
Maybe the 2nd sentence spoke about how hard life became for the author that even food turned to be harsh in their throats, like bran.