To a Rabbinic Jew who doesn't hold Jesus as the Messiah, the incarnation is an a shock to him or her to put it mildly and may not be comfortable to use the term person.
When Christians say that God is a person, we so not mean that he has a human body, by person we mean one who is a able to think, reason, plan, interact and have emotion.
Father, Son and Spirit are individual distinct persons in the Godhead who interact with in another and communicate with one another.
Jesus, the Son, Second Person in the Trinity, took on human flesh in the first century but did NOT ceased to be God in heaven. He veiled his glory but did not cease being God in heaven.
Despite not being human until he came to earth, he nevertheless could be classified as a person in light of what I have just said above.
It comes down to defining our terms, especially with respect to Yitzhak Shapiras recent videos on the Trinity which I am going to comment on in the future and of course Sam Shamoun has observed the usage of manifestation by Shapira which needs to be examined.