No funny notion of ⊂ at all, it is about how you construct ℚ, for example. You construct Q(ℤ), show that ℤ is embedded in Q(ℤ) via a monomorphism I, and then define ℚ as (Q(ℤ) \ I(ℤ)) ∪ ℤ. For example. You are right that nobody cares about how this works in detail. It does assume though some notion of ℚ and ℤ as collections, but not necessarily sets from ZFC.