Yeah, that's how I understood it. E.g one might write

  (a+b)^2 != a^2 + b^2

To mean that in general the equality doesn't hold. Despite exceptions like a=b=0

Strictly you should write something like

  ¬[∀ a,b  (a+b)^2 != a^2 + b^2]

But shorthand and abuse of notation are hardly rare

No, that can be the case also for mathematical entities for which you can know the values, not just for "unknown variables".