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Why is PhD read as /piːeɪtʃˈdiː/ (from Oxford Dictionary) and not, for example, like /fˈdiː/ , while diagraph ph is read as /f/ in Latin and Greek words? Why do we write Ph if not to represent the /f/ sound?

There are questions about writing (like this) but not pronouncing.

Because it is an initialism so you read out each letter ("DVD" is pronounced "dee-vee-dee", not "dvid"; "US" is pronounced "you-ess", not "uhs"). Your proposed pronunciation could be used were it an acronym.

PhD (or Ph. D.) is a bit of a frozen expression or idiom. The expression doesn't abbreviate the English phrase "Doctor of Philosophy". If it did, then it be something like "DP" or "DoP". Instead, PhD retains the structure of the medieval Latin Philosophiae Doctor, which dates from the 17th century.

As to why the Latin abbreviation for "Philosophiae" was "Ph" rather than just "P"? "Philosophia" was a word borrowed into Latin from the Greek, and in Greek the word is spelled "φιλοσοφία", the first letter being φ. In Greek that's a single letter representing an aspirated π, and is transliterated into Latin as ph.

Since the abbreviation PhD does not match up with the English phrase it supposedly abbreviates, the pronunciation of the abbreviation has diverged from the pronunciation of the phrase.