There is no shred of evidence nor any reason to believe Dhul Qarnayan to be Alexander, zero, zilch, nada.
Other than the fact that the DQ narrative is pretty much identical to the Syraic Alexander legend, which is an evolution of the older Alexander romance. The fact that 'two horned' matched the iconography of Alexander. Hadith noting that DQ built a city formerly known as Alexandria, etc.
Al-Kahf is commentary on 3 religious stories common to the time period, although not Biblical ones. The 7 sleepers of Ephesus was also a common story in the Late Antique Roman Empire
Cyrus The Great on the other hand, as I already said, has a massive reputation with all three Abrahamic religions (regarded by Jews too as one of the messiahs of the Tanakh). Everything about Cyrus The Great perfectly aligns to it, historically and textually.
If you want Tanakh proof, start with Isaiah which highly praises him again and again and again. Then move your way through the Nevi'im, where again he is typified as the ultimate messianic archetype (aside from David and Moses respectively).
As pointed out, Alexander was also commonly used in Christian stories. Emperor Heraclius ('the new Alexander') was also using Alexander to enhance his reputation at the same time by means of imperial propaganda.
It's not only exactly the same story, it's the same story at the same time.
Claiming him to be Alexander is a modern thing though, it's an orientalist myth largely originated in the 19th century.
And they the Jews question you concerning Dhū’l-Qarnayn whose name was Alexander; he was not a prophet. Say ‘I shall recite relate to you a mention an account of him’ of his affair.
Tafsir al-Jalalayn (15th C)
Hadith and tafsir making a connection to Alexander despite it being a 'modern orientalist myth' with 'not a shred of evidence" seems a bit incongruous, no?
The identity of Qarnayan is never addressed in any Hadith, so you will have to either look at what makes the most logical sense (Cyrus, again and again) or admit that you don't know because it's way too vague otherwise. Qarnayan is only mentioned in around 20 Ayah in Surah 18, it's not explicit enough to any degree to justify your Alexander myth - and at that, the Alexander myth doesn't date back very far according to primary sources. There is very direct and clear continuity with Cyrus, but zero with Alexander.
And aside from that there aren't only two options to who he may have been (setting aside from that fact that Surah al-Kahf is an apocalyptic Surah dealing with parallelisms of past matters to future eschatological matters, it's a very unique Surah for a lot of reasons...)
The most compelling evidence is the fact that it is the same story that we know existed in the same time and the same place. Not 'slightly resembles', the same. Read the earlier linked article if you would like to see the evidence