But the integers are a subset of the rationals, which are a subset of the reals, which are a subset of the complex numbers. Looking only at the objects and not their operations 1 (integer) = 1 (rational) = 1 (real) = 1 (complex). Moreover, when we do account for the operations, we also see that 1 + 1 = 2 and 1 * 1 = 1 in every one of those systems. This isn't just a coincidence, of course; it's by design.

However, the way you arrive at 1 + 1 = 2 is not the same (though I suppose you could short-circuit the algorithm). Rational addition requires finding a common denominator, while integer addition doesn't. They achieve the same result when the inputs are integers, and again this is by design, but the process isn't the same. Ditto real addition vs. rational and complex addition vs. real.

In higher-level mathematics, the operations on the objects become definitional. We don't look at just a set of things, we look at a set of things and the set of operations upon those things. Thus "1 with integer addition and integer multiplication" becomes the object under consideration (even if it's just contextually understood) instead of simply 1. This is why they don't satisfy higher-level notions of equivalence, even if they intentionally do satisfy simple equality as taught in grade school.

Of course, the entire point of the submitted paper is to examine this in detail.