But it's not completely different and a "≃" would not mean the same thing, in fact it would weaken the statement.

e^x ≃ 1 + x + O(x^2) would only assert that lim (x->0) (e^x)/(1+x) = 1.

However "e^x = 1 + x + O(x^2)" means that for some function r(x) belonging to the set O(x^2), e^x is exactly equal to 1 + x + r(x). Another way to rewrite that equation that eliminates the "abuse of notation" would be:

    e^x − (1 + x) ∈ O(x^2)

The particular r(x) in O(x^2) which makes it strictly equal is being left out, that's true, and usually it's left out for brevity or practical reasons or even because it's not even be known what r(x) is... but nevertheless it is not an asymptotic equation or an approximation, it is exactly equal to the value on the right hand side for some particular r(x) the exact details of which are being omitted for one reason or another.