Short answers, pulled from the story.
Welsh mathematician Robert Recorde invented the equals sign and first published it in The Whetstone of Witte in 1557, one year before his death. He called his two parallel lines "Gemowe lines", from the Latin for twin, explaining that no two things could be more equal.
Recorde's symbol went unused in print for sixty-one years after its 1557 introduction, reappearing only in 1618 in an appendix to John Napier's Descriptio. Broader recognition in England came in 1631, and widespread European adoption followed through the influence of Isaac Newton and Gottfried Leibniz.
The substitution property states that if a equals b, any property of a is also a property of b. It traces to Gottfried Leibniz's Discourse on Metaphysics around 1686 and is often called Leibniz's Law. Russell and Whitehead formally introduced it into symbolic logic in Principia Mathematica between 1910 and 1913.
In Zermelo-Fraenkel set theory, two sets are defined to be equal if they have exactly the same members. This principle is called the axiom of extensionality.
An equation may be true only for specific values of a variable, while an identity holds for every value in a given domain. The triple-bar notation sometimes used to mark identities was introduced by Bernhard Riemann in his 1857 lectures on elliptic functions.
Andrey Kolmogorov proposed restructuring geometry courses through set theory, in which a figure is a set of points and can only be equal to itself. This framework made congruent the standard school term for figures of the same shape and size, replacing the older usage of equal, and gained widespread adoption around the 1960s.