There are generally two schools of thought in WRO, one that resembles the Catholic Church (Antiochian WRO) and one that doesn't (ROCOR) so it's byzantinized, guesses what pre-schism practice looked like, or imitates Anglicanism (as long as it's not Roman).
The Council of Baltimore, just a local council, of the Catholic Church in the 1880s set the Roman Rite standard for clerical attire in America up through the days of the late, great Cardinal Spellman: on church property, cassock and biretta; off church property/street clothes, black suit and Roman collar (usually the "notch" collar but in early days you saw the Anglican-style band collar too). And when most men wore fedoras, a black one was authorized as street wear in the Archdiocese of New York and I'm sure elsewhere in America.
At least informally the Orthodox did the same. I've met older Greek-American priests who looked just like traditional Episcopal priests: black suit and white band collar. OCA (especially older ones) and Antiochian priests on the street look like Roman Rite Catholic priests: the modern variant of the black suit, which is black trousers and black shirt with white tab collar. Great! To the average American they're clearly priests, not Protestant ministers. Younger OCA priests might look more Russian, in Russian cassocks and beards.
Because of the Antiochians' Catholic-friendly approach to WRO, their WRO archpriests have used "Monsignor" and the corresponding cassock.
(So did ACROD for many years; they were pushed out of the Catholic Church for no good reason, not really about theology. I think they saw the practice as more "naš," "ours," than Catholic because they'd used it for centuries. Their reason to exist was well-meant and understandable: they wanted things to stay the same. So for a long time, they kept it.
Also, European Catholic bishops have used "Monsignor," so traditionally the Catholic Church referred to Orthodox bishops that way too, both recognizing their orders and slyly claiming authority over them: the Archbishop of Canterbury was called Dr. Fisher but the Patriarch of Moscow was called Msgr. Alexis, which I admit is a bit rude.)