The notation is there to help us explain and communicate what are abstract thoughts in our heads. As such, if people understand what we are thinking/saying, it is close enough. And, being exact would only make things longer and more confusing.
I agree that the actual example you give isn't an abuse: we all know that a mapping from $R^2$ to $R^1$ has two 'variables', which are the 'coordinates in $R^2$, or if you like, or the elements of the tuple in $R^1xR^1$; So, if $z=x^2+xy+y^2$, we all know what we mean.
I suspect part of your concern comes from computer science. There, tuples must be explicit and track for software to work: in Python, a tuple is declared like $a=(1,2,3)$. But even at modern AI sophistication, software can't infer stuff. Technically this is because all software requires what is called an $LR1$ grammar (which means it is sufficiently unambiguous), which guarantees there is zero ambiguity. As a consequence, computer languages often have exact but confusing syntax because they slavishly follow grammar/notation rules with zero abuse: in Python, the following also creates a tuple of length 1: $$a=1,$$ which looks, for all the world, wrong.
I often think that how you know that you understand a concept when the notation is 'abused' (or perhaps confused), and symbols get repeated -
but have different meanings you see immediately For example, if you were dealing with basis vectors over multiple dimensions that were indexed, and also exponentiating, An expression like this looks perfectly sensible:
$$z=\sum e_i^ {e^{{\frac{ix^2}{\sigma^2}}}}$$
even if it isn't clearly sensible, given the context and our proper understanding of what is going on, we might all know the $e$ downstairs is a basis vector (though a hat over it would help), and that the $e$ upstairs is 2.718, and - given the context - that the $i$ upstairs is likely = $\sqrt{-1}$, and not the index $i$ from the $e$ at the bottom. There is ample possibility for confusion by someone who doesn't grasp the math, but those who get it can untangle the repeated symbols (the $e$'s and $i$'s in my example).
A sort of extreme version of this is the Einstein notation convention, where you drop the $\sum$ and just 'intuit', based on the logic, that it is to be summed over all necessary subscripts...so $x=a_j x_ij b_j$ has implied summation - but we don't need to say it.
There are endless examples. We say that a space X is measurable, but really that requires a tuple like $(X, \mathbf X, \mu)$, which gives the underlying space, the $\sigma$-algebra, and the measure. But, if we know the $\sigma$-algebra then we could figure out the space; and, really, if we know the measure and topology, we can figure the $\sigma$-algebra. So it would be more accurate to talk about the measurable space $\mu$, but we all 'know what we mean'.
Abuse of notation? Sometimes, I suppose. But notation needn't be slavish, and should make it simple and clear - and we can 'fill in the blanks'. that is what still separate humans from machines.