The question uses the technical term "relation", which has a formal meaning in mathematics. These days, we can use the language of relations to say that numeric equality is an example of an "equivalence relation", meaning that it fulfils certain conditions such as "transitivity". Relations can be modeled in set theory, if we want to. There are books about this.
And yet I have many cousins and other relations who are not mathematical objects.
There is a group of kids over by the bus stop, but they are not endowed with an associative binary operation having two-sided identity and inverses.
If I tell you my kitchen door is open, I'm not making a statement about its status in a topological space.
I can also say things like "those candidates are equally disreputable", without meaning to imply that there is a common numerical scale of reputation according to which the candidates have equal placement. What I am trying to say is that they are both bad and I don't care to distinguish between them in the matter of reputation.
When we speak of people being equal in a political sense, we are not making a mathematical assertion. We're saying that the people are on the same basis with respect to their participation in civic society. While that could be dressed up in mathematical language, people can perfectly well talk about political equality without having any mathematical knowledge at all.
Mathematical words are frequently borrowed from normal speech, so that the words chosen have some connection to the matter under investigation. In the case of "equal", this is so far in the past that we can't pinpoint an origin, but there are many more recent developments that are well-documented. We have come up with generalizations and distinctions and axioms, because in a mathematical context it pays to be precise. We've realized that "equality", once thought to be so basic that it doesn't need explanation, actually does demand more formal attention. That doesn't stop a term from being used in a non-mathematical context, where it doesn't carry the same rigorous definition.