[zozbot123]

Yup, you can define distinctness of constructive real numbers as two algorithms returning rationals that will differ by more than the approximation error ε they were given as input, for some arbitrarily small ε - but equality involves proving that two algorithms will never be apart in this way - and this cannot be done in general. As you say, even comparison with zero is problematic. So instead of equivalence relations, you need to define the apartness relation - the notion that a supposed equivalence between two such numerical algorithms can be constructively refuted. This is nothing new - it's a well-known part of constructive math, and is also the sensible way to formalize numerical analysis.