Abelian category equivalent to a non-abelian category

Here is a manifestly invariant definition of an abelian category $\mathcal{C}$. It is a category with finite limits and colimits such that:

1. (It is pointed) the map from the initial to the final object is an isomorphism; we denote by 0 any object which is both initial and final.
2. (It is semiadditive) the map $X \amalg Y \to X\times Y$, given on $X$ by components $(\mathrm{id}_X, X \to 0 \to Y)$ and on $Y$ by components $(Y \to 0\to X, \mathrm{id}_Y)$, is an equivalence. We denote by $X\oplus Y$ the coproduct or product, identified as above. This equivalence produces an abelian monoid structure on all hom sets, where addition arises from $X \to X\oplus X \stackrel{f\times g}{\to} Y\oplus Y \to Y$.
3. (It is additive) the shearing map $X\oplus X \to X\oplus X$, given by adding the identity map to the projection onto the first component followed by inclusion, is an equivalence. Equivalently, each hom-monoid has the property that it is group-like.
4. (first isomorphism theorem) if $f: A \to B$ is arbitrary, then the map $A/\mathrm{ker}(f) \to \mathrm{ker}(B \to B/A)$ is an isomorphism.

Being an abelian category is a property not structure.

What you were told is wrong, for we have the following:

Proposition. If two categories are equivalent and one of them is abelian, then so is the other.

A proof (and some related results) can be found in Satz 16.2.4 in H. Schubert, Kategorien II, Springer, 1970 (likewise in the English version https://www.amazon.com/Categories-Horst-Schubert/dp/3642653669, under the same numbering).