I know that by his time in the middle to late second century, Rome had given up much of it’s earlier suspicion of Greek culture and Hellenism was fully integrated into the culture of the western Empire, but Latin was still the primary language of Roman society and I am assuming that Marcus Aurelius, given his upbringing, parentage and the era in which he lived, spoke Latin natively and would have learned Greek only as a second language.
From what I know his Meditationes was a private diary, not intended to be published or seen by anyone, so why would he write his private diary in [Koiné] Greek, unless I am incorrect and his first language was Greek? If so, was that a common thing amongst the Roman upper class, or something unique to him or his family?
First, elite Romans learned Greek as part of their education. Enslaved Greek-speaking pedagogues became common once Greece was conquered (like the sack of Corinth in 146 BCE), and we are talking about a period centuries after that (121-180 CE for Marcus's life), where educational practices would have been well-established. I also want to emphasize that the eastern half of the Roman empire used Greek as an official language; plenty of Roman citizens left behind writing in Greek. So, it is not unusual, but expected, for the emperor himself to be well-versed in Greek. In particular, Greek was important for learning rhetoric and philosophy in Marcus's era, which I will continue on below.
Second, why write a private journal in Greek? This comes down to genre. We have some surviving Latin letters between Marcus and his teacher Fronto (see Haines). For at least some correspondence, and probably most other writing, Marcus used Latin. He likely chose Greek for the Meditations because his personal musings in the Meditations are philosophical in nature, and Greek was the language of philosophy. The Meditations are not a "diary" like how we think of them today--it says almost nothing about what Marcus is actually doing day-to-day. Scholars think Marcus wrote the Meditations toward the end of his life, while on campaign against Germanic tribes, but details of these campaigns are largely absent from the work. Instead, he records philosophical musings, such as ideas on mortality and morality. Some of his work draws on the philosopher Epictetus, who also wrote in Greek, and likely other Greek philosophers whose works don't survive.
Additionally, Marcus was operating within what is now termed the Second Sophistic period, a resurgence of Greek declamation and philosophy (ca. 60-230 CE). "Sophists" are typically limited to rhetors, but the period is often used by classicists to describe a wider resurgence in Greek literature (all written under Roman rule) during this period, including authors like Lucian, Aelius Aristides, and Galen (who happened to be Marcus's personal physician).
Although Marcus is writing for himself, he is writing within context of resurgent Greek literature. His choice of Greek for philosophy would have seemed natural to him, and allowed him to quote and engage with the works of other philosophers seamlessly. The private nature of the Meditations could also be why he chose Greek; had he intended this for wider circulation, he may have used Latin. To sum up, Marcus Aurelius would have learned Greek from a young age and his choice to use it in his personal writing fits his education, status, genre, and historical context.
Bibliography:
C. R. Haines, The Correspondence of Marcus Cornelius Fronto with Marcus Aurelius Antoninus (Loeb, 1919–20)
D.A. Russell, ed., Antonine Literature (Oxford University Press, 1990)
R.B. Rutherford, "Aurelius, Marcus, Meditations." Oxford Classical Dictionary.
J. Sellars, Marcus Aurelius (Taylor & Francis, 2021)
T. Whitmarsh, The Second Sophistic (Oxford University Press, 2005)