"Pope/bishop Alexander" is a fake, he really matches Trajan.
Some extra biographies details matches evidences from their biographies:
-- Alexander: name Alexander. -- Trajan: Trajan is said to have been "thinking aloud of Alexander [the Great] at Charax". He "sent the Senate a laurelled letter declaring the war to be at a close and bemoaning that he was too old to go on any further and repeat the conquests of Alexander the Great." "as Trajan left the Persian Gulf for Babylon – where he intended to offer sacrifice to Alexander in the house where he had died in 323 BC". "greatest military expansion" / extended empire [like Alexander the Great]. fine/succesful soldier. [Trojanu could link with Paris/Alexander of Troy??] [Legate Tiberius Julius Alexander Julianus.]
-- Alexander: Ruled 105/106/107/108/109-115/116/119. "in reign of Hadrian" / "ruled in reign of Trajan" / "under Trajan/Hadrian". -- Trajan: reigned 98-117. Born 53.
-- Alexander: crown. sceptre. -- Trajan: Emperor.
-- Alexander -- tradition says he was a Martyr by decapitation, which is said to be improbable (other early source says nothing about his Martyrdom). [Disputed whether martyr Alexander same as pope Alexander.] -- Trajan -- by inversion: 11000 dead in Trajan's gladatorial festival. [Semi subterranean cemetery of martyr Alexander, Eventius, and Theodolus?]
-- Alexander: "Saint". shown with a halo/aura. -- Trajan: deified.