Given two parallel morphisms f,g:XtoY in some category mathcalC, let us consider the category mathcalEf,g :
Objects : all pairs (E,e), where E is an object of mathcalC and e:EtoX is a morphism in mathcalC such that fcirce=gcirce,
Morphisms : from (E′,e′) to (E,e) are just morphisms varphi:E′toE in mathcalC such that e′=ecircvarphi.
Now an equaliser of f,g is just a final object in the category mathcalEf,g. Final objects in any category are unique (provided they exist), up to a unique morphism; we may then talk of "the" equaliser of f,g.
When mathcalC is the category of sets, the equaliser always exists (and is therefore uniquely unique); as you say, it is the largest subset of the common source where the two maps coincide.
It is fine to think of it as a "maximal" object in mathcalE=mathcalEf,g, but one must realise that it is also a "minimal" object in the opposite category mathcalEcirc in the sense of being an initial object therein.
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