Both are named Alexander: Trajan has 3 associations with the name Alexander: (1) Trajan was identified with Alexander the Great (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".) (2) legate Tiberius Julius Alexander Julianus? (3) semi-subterranean cemetery of martyr Alexander, Eventius, and Theodolus? ("Trajan was the emperor, he martyred Alexander." So they falsely identified the martyr with the emperor he was martyred by/under, like they did with Peter and Nero (see the 1st pope).)
(The meaning of the name Alexander "helper/defender of mankind" may also relate to Trajan?)
Dates match: Pope Alexander 1 reigned 105/106/107/108/109-115/116/119 (compiled from six different official popes lists). Emperor Trajan reigned 98-117. Both dates coincide 105-116/117 (the whole of pope Alexander's reign), same end date 116/117.
Both are pontiffex maximus: Alexander 1 is bishop / "pope" / "pontiffex maximus" / "supreme pontiff" (Annuario Pontificio, Liber Pontificalis). Trajan was pontiffex maximus (the emperors from Augustus to Gratian were pontiffex maximus). [The argument that the titles pontiffex maximus and pope were not used by/for bishops of Rome until later (mid-5th & mid-6th centuries) and so is an anachronism is not necessarily valid because bishop/pope/pontiff are still related.]
Both are in Rome.
With both the popes & emperors before and after match all in order. 1 Peter (Nero) = Nero (Peter) 2 Linus = Vespasian 3 Cletus = Titus 4 Clement 1 = Domitian (Clemens) 5 Evaristus = Nerva 6 Alexander 1 = Trajan (Alexander) 7 Sixtus 1 = Trajan? or Hadrian? 8 Telesphorus = Hadrian? 9 Hyginus (Graecvs) = Hadrianus (Graeculus) 10 Pius 1 = Antoninus Pius etc. See each separate pope & emperor chapter for matches details.
Same number: Alexander is 6th from Peter. Trajan is 6th from Nero excluding the 3 "usurupers".
Pope Alexander a "saint", shown with a halo/aura. Emperor Trajan deified.
Pope Alexander: crown, sceptre. Emperor Trajan: emperor.
Pope 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: 3rd persecution of christians under Trajan & Hadrian. Semi-subterranean cemetery of martyr Alexander, Eventius, and Theodolus? "Trajan was the emperor, he martyred Alexander."
There is no solid contemporary proof that any pope Alexander 1 ever existed. "Many popes in the first three centuries of the Christian era are obscure figures". "St Irenaeus bishop of Lyons of the 2nd century mentions various historic Roman bishops", "Irenaeus mentions Alexander". But does he say Alexander was bishop of Rome? It is disputed whether martyr Alexander is the same as pope Alexander. Even if Alexander were bishop, they still could have conflated the martyr with the emperor he was martyred under like they did with Peter & Nero (see 1st pope). The date they first came up with the popes list is not certain, maybe they first came up with the pope/emperor correspondences even as early as Irenaeus?