I'm struggling to understand how the ratio of inputs to outputs relates to linearity.
Determine if the function is linear or nonlinear.
If I am not mistake, this has 2 inputs, 1 output: $f(x) = 2x_1 + 3x_2$
But this function has 2 inputs, 2 outputs: $f(x) = [3x_1 + 2x_2, -4x_1]$
I tried superposition and homogeneity but couldn't get anything sensible.
The number of inputs, number of outputs, or their ratio has absolutely nothing to do with linearity.
Apart from anything else, "the number of inputs" or outputs is not a clearly defined concept. If you have, as in your second example, $$f(x_1,x_2)=(3x_1+2x_2,\,-4x_1)\ ,$$ you could say that $f$ has two (numerical) inputs, or one (vector) input. It's a matter of language, not really a question of mathematics. Another example: is $$\pmatrix{1&2\cr3&4\cr}$$ four numbers, or is it one matrix? (Or two column vectors maybe?)
For the record, both your examples are linear. But it has nothing to do with the "number of inputs/outputs".