dimensional analysis - Why can we only add or subtract things of the same dimension (meters/second, newtons, etc)?

Dimension analysis is a nice tool to create functions using physics dimensions that are desirable for our lives. I know, as an axiom, of sorts to me, that addition and subtraction must conserve units, and thus are actually relatively uncommon in physics equations.

Now, for why that is, I would tell someone it's the same reason why you only can add terms with the same qualities as that term in math as well (such as $x^2$ only being able to be added with scalar multiples of $x^2$, the same applying to $y^5$ or $e^{-2x}\cosh(3x)$, for example). Except in the case of inputs, we have... (and here is where my explanation gets shady), vector units or magnitudes with magnitude 1 of things such as $\text{meters/second}$, $\text{Newtons}$, $\text{Tesla}$, $\text{Joules/Coulomb}$ etc.

However, I can't seem to necessarily explain why I couldn't say, add $x^2$ to $x$ or a velocity to a force other than my body simply not letting me out of habit, and my only explanation is: "You just can't", or "It'd look bizarre."

I need a better explanation for this. Could someone enlighten me? My two questions:

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  Why does adding and subtraction break down at things with different dimensions?

  
  


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  Why does multiplication/division allow for it?