If you don't have (extensional) power sets either, then you'll have to use setoids (in which case, perhaps you'd do better to change your terminology as described there). Alternatively, don't worry about any of this and just include the existence of quotient objects in Set as an axiom (the axiom of quotient sets, which you have if you assume that is a pretopos or a Grothendieck topos as given by Giraud's axioms).
In any case, the element of that comes from the element of may be denoted , or simply if is understood, or simply if there will be no confusion as to which set it is an element of. This is called the equivalence class of with respect to ; the term ‘class’ here is an old word for ‘set’ (in the sense of ‘subset’) and refers to the definition (1) above, where is literally the set .
Quotient sets in Set generalise to quotient objects in other categories. In particular, an exact category is a regular category in which every congruence on every object has an effective quotient object.