This is not really in the spirit of the examples you give but it is at least a set of purely categorical properties. 

> **Proposition:** A category $C$ is the category of models of a <a href="https://ncatlab.org/nlab/show/Lawvere+theory">Lawvere theory</a> iff there exists an object $P$ (the free object on one generator) such that the functor $U = \text{Hom}(P, -) : C \to \text{Set}$ preserves <a href="https://ncatlab.org/nlab/show/sifted+colimit">sifted colimits</a>, is conservative, and has a left adjoint $F : \text{Set} \to C$. 

*Sketch.* By the <a href="https://ncatlab.org/nlab/show/monadicity+theorem">crude monadicity theorem</a>, the adjunction $F \vdash U$ is monadic, so $C$ is the category of algebras over the monad $UF : \text{Set} \to \text{Set}$. Since $UF$ preserves sifted colimits, and in particular preserves filtered colimits, it is a <a href="https://ncatlab.org/nlab/show/finitary+monad">finitary monad</a>, and these are known to be equivalent to (the monads induced by) Lawvere theories, with the equivalence also sending categories of algebras to categories of models. $\Box$

This set of categorical properties could be replaced by other ones, e.g. we could use a characterization of <a href="http://www.numdam.org/article/CTGDC_1994__35_4_351_0.pdf">categories monadic over $\text{Set}$</a> as the <a href="https://ncatlab.org/nlab/show/exact+category">Barr exact</a> categories with a compact projective generator $P$. But 1) the compact projective generator of the category of models of a Lawvere theory given by the free object on one generator always satisfies the stronger property above that $\text{Hom}(P, -)$ preserves sifted, not just filtered, colimits, and 2) because reflexive coequalizers are sifted colimits, this lets us apply the crude monadicity theorem without needing to know anything else about the behavior of coequalizers. So I think this characterization is a bit simpler. 

Now $\text{Grp}$ can be isolated as the category satisfying the above properties such that the monad $UF$ is the free group monad. I don't expect this to be particularly satisfying but it does at least establish that $\text{Grp}$ can be isolated among all categories via categorical properties.