I feel like that example counters your claim in two different ways.
First, (1, 2) only makes sense if you assume that x is first and y is second, or in other words that they correspond to X_1 and X_2 for some vector X. This is probably a reasonable assumption for x and y, but what if you have some other arbitrary choice of variables?
a ρ + 2 = 0
Then the only unambiguous way to write a solution is with explicit labels: a = 1, ρ = -2.
Second, another way to “solve” the equation “x y + 2 = 0” is: “y = -2/x, for any x ∈ ℝ \ {0}”. You could argue that this is also a different kind of “equation” than the one you started with: the reason it can be seen as a solution is not (just) that it’s simpler than the original equation, but that it provides an algorithm to enumerate the set of individual solutions, as well as the set of solutions given some proposed x value. (That is, a set with one or zero members depending on whether x = 0.) However, even if it is a different kind of object, there’s no way to represent it in standard notation that doesn’t conflate ‘questions’ with ‘answers’. If you really want to use tuples, you could go for “{(x, -2/x) | x ∈ ℝ \ {0}}”, which avoids the equals sign – but the ∈ is playing a similar role, an algorithm (enumerate all members of this set) disguised as a test (is this value a member of the set?).
edit: Upon further reflection, I might actually just be expressing violent agreement with the point you were trying to make. shrug