I am interested if the phonetic similarity between the English "baby" and the Chinese "宝贝" (baobei) is purely coincidental.
Interesting, but this is a coincidence.
寶貝 comes from the meaning rare/precious seashells; this usage is attested at least since the Han dynasty. As a term of endearment, this started appearing as early as in the novel Dream of the Red Chamber, written in the 18th century.
They are not etymologically related as other posters have noted, but it might not be a pure coincidence.
Many people have noted that words like "mama" and "papa" are similar across language families and suggested this is because they imitate the types of babbling sounds infants make. See: https://en.wikipedia.org/wiki/Mama_and_papa
It seems plausible that "baby" could also fall within this category (i.e., repeated simple syllables.)
宝贝 means "treasure" in Chinese. When use 宝贝 for babies it means babies are the treasure of families.