There is a standard mathematical definition for an "equivalence relation". Are you familiar with it?
There can be several different equivalence relations defined on the same set of things. Two things related by a certain equivalence relation R are said to be "equal with respect to R". In a given context it may be clear what equivalence relation is being used. For example, in a text book discussing elementary algebra with the real numbers, the equivalence relation is understood to be the one we learn in elementary arithmetic. So the book won't bother to say two numbers are "equal with respect to the equivalence relation on the real numbers". It will just say the numbers are "equal" or use the abbreviation "=".