But think going deeper into this is quite fun
Your post goes to the point at the heart of philsophical number theory.
What does equality mean ?
Yup – you got functinal equivalence, isomorphism, and temporary assignment of values.
But I think you could prove – that all these types of equality – are “instances” of “different implementations” of “equivalence.
They are no more equivalent than 1 = 1 is equivalent.
I.e. 1 = 1 means I think we can define a bijective “counting function” that proves there’s the “same number” of “elemetns” in the “sets”
I think (not sure) – if you define – counting fucntion / same number / elements / sets differently – you get the differing definitions of equivalence you enumerate.
The interesting thing for me is that 1 = 1 is defined clear in 4 of peano’s axioms
https://en.wikipedia.org/wiki/Peano_axioms#Formulation
And you could mentally – try to develop different (and potentially) – more powerful notions of “equivalence” – with differing axioms
A final point… the prevalence of several “similar” concepts of equivalence in computer science – may point to an underlying “platonic idea” of equivalence – that either exists dormant in the world awaiting for us to discover it; or is a useful “technologocial” construct – that has accelerated “progress”