Short answers, pulled from the story.
Robert Recorde published The Whetstone of Witte in 1557 to introduce the equals sign for the first time. He chose two parallel lines called Gemowe lines from the Latin word twin because he believed no two things could be more equal.
Ernst Zermelo first explicitly formalized set equality as part of his theory published in 1908. Abraham Fraenkel later contributed to what became known as Zermelo-Fraenkel set theory during the foundational crisis of mathematics at the turn of the twentieth century.
Giuseppe Peano explicitly stated these properties as fundamental in 1889. Ancient Greek thinkers understood reflexivity symmetry and transitivity long before they wrote them down as rules around 350 BC when Aristotle defined quantity in terms of a primitive notion of equality.
Richardson proved that equality between real numbers defined by expressions involving logarithms is undecidable. Computers represent real numbers as floating-point values with limited significant digits which creates fuzzy ranges rather than exact values due to physical constraints.
The substitution property remained informal until Gottfried Leibniz formulated it in Discourse on Metaphysics in 1686. Leibniz argued that no two distinct things can have all properties in common while function application became common practice in algebra since at least the time of Diophantus.