Why would the Ancient of Days and the Son of Man as appearing in Daniel be the same person? God the Father is not a human being, so I would never use the word 'person' for Him in this day and age (that would cause confusion given the definition of 'person' in any English dictionary). The ancient of Days and the Son of Man in this passage obviously are two distinct beings/identities though - and that's fully Biblical; one can see the same pattern in Psalm 110:1, and several others ..
The word prosopon used by the Fathers to refer to the three Persons of the Holy Trinity is commonly and correctly translated as person, for we worship One God, the Father, Son and Holy Ghost. The All Holy, Undivided and and Life Giving Trinity consists of three coequal, coeternal and uncreated persons, ever one God, united eternally in a union of perfect love, with the Son, begotten of the Father before all ages, and the Holy Spirit, eternally proceeding from the Father, together sharing in the Divine Essence of the unoriginal Father, with each Prosopon having His own Hypostasis, or underlying reality. In the case of our Lord, God and Savior Jesus Christ, in His Incarnation our human nature became hypostatically united to HIs uncreated Divine nature, without change, confusion, separation or division.
Person is the most appropriate word to use even if we consider the ancient use of Prosopon meaning Mask or Face, since if we combine Prosopon with Hypostasis, it adds that element that is implied in the English word Person that the word Prosopon would on its own have in deep antiquity lacked.
It is advisable to stick to the Patristic language and not risk using words such as “beings” or “identities” with regards to the Holy Trinity, as these could have the effect of confusing the relationship between the three Divine Persons.
Since your posting in the Eastern Orthodox subforum causes me to suppose you have an interest in learning about Eastern Orthodox theology, I would suggest you read
The Orthodox Way by Metropolitan Kallistos Ware for its excellent explanation of the doctrine of the Trinity. For more of a deep dive,
The Exact Exposition of the Orthodox Faith by St. John of Damascus and
De Incarnatione by St. Athanasius, if you are able to read Patristic writers; if you struggle in this department, a more recent work is
Orthodox Dogmatic Theology by Protopresbyter Michael Pomazansky, memory eternal, translated by Fr. Seraphim Rose, memory eternal; this is the best contemporary work of Orthodox dogmatic theology I have seen.