Ancient Greek did not run short of the name Eucleides, and confusing the Athenian archon with either of two much better remembered men who shared it is an easy mistake to make. Euclid of Megara, active a generation after the archonship, founded the Megarian school of philosophy, studied under Socrates, and is remembered chiefly for logical paradoxes and for sheltering Plato and other Socratics in Megara after Socrates's execution made Athens unsafe for his followers; he held no civic office and had nothing to do with the alphabet. Euclid of Alexandria, active roughly a century later still, is the mathematician whose Elements systematized geometry for the ancient world and whose name survives today attached to Euclidean geometry generally; he too held no Athenian office and worked in an entirely different city, field and century.
The Athenian archon shares only the name with both of them, and the confusion is compounded by "Eucleides" and "Euclid" being simply two English renderings of the identical Greek name, so a loosely sourced reference can easily attach the archonship, the philosophy, or the mathematics to the wrong man. What can be said with confidence about the archon is narrower than what is known of either namesake: he held the office of eponymous archon at Athens for 403/402 BCE, the year restored Athenian democracy is dated from, and in that capacity presided over the decree adopting the Ionic alphabet. No philosophical school, no mathematical treatise and no other recorded act of his has come down to later readers. He is, in the fullest sense, a man known by his office and by one decision made in it.