Abelian varieties have tropicalizations, but the picture is not exactly as you guess.
Let me begin with the case of elliptic curves. So let $A$ be an elliptic curve over a non-archimedean field $K$, let $j$ be its $j$-invariant. Two possibilities arise.
If $|j(A)|\leq 1$, then, possibly after enlarging $K$, it is possible to find a Weierstrass equation for $A$ whose reduction modulo the maximal ideal of the valuation ring of $K$ still defines an elliptic curve. One says that $A$ has good reduction. In some sense, this is the best of all possible cases from the point of view of arithmetic geometry, but unfortunately it escapes the world of tropical geometry.
So assume that $|j(A)|>1$. In this case, whatever Weierstrass equation you choose for $A$, whatever extension of $K$ you take, you won't be able to find one which reduces to the equation of an elliptic curve. However, it will eventually reduce to the equation of a nodal cubic (eg $y^2=x^2(x+1)$). Then, as was observed by Tate, and this is really the beginning of non-archimedean analytic geometry, $A(K)$ can be understood as the quotient of the multiplicative group $K^\times$ by a discrete subgroup of the form $q^{\mathbf Z}$, for some $q\in K$ such that $0<|q|<1$. In his paper « A review of non-Archimedean elliptic functions », Tate has given a wonderful account of this picture in the book Elliptic curves, modular forms and Fermat’s last theorem (J. Coates and S.-T. Yau (eds.) (1995), 162–184).
Tate's point of view is that the classical 19th-century formulas on complex elliptic functions (ie meromorphic functions on $\mathbf C$ with two periods $1$ and $\tau$ — of positive imaginary part), especially Jacobi's, still have some meaning in this framework provided one first mods out by the subgroup generated by $1$ and writes everything in terms of $e^{2\pi i z}$. That is $\mathbf C/(\mathbf Z+\mathbf Z\tau)\simeq \mathbf C^*/q^{\mathbf Z}$, with $q=e^{2\pi i \tau}$.
In rigid analytic geometry, this is interpreted as saying that our elliptic curve $A$ (such that $|j(A)|<1$, recall) has some « $K$-analytic uniformization » by non-archimedean elliptic functions. But now $K^*$ tropicalizes to the real numbers, by the familiar map $z\mapsto \log|z|$, so that $A(K)$ tropicalizes to the real torus $\mathbf R/\mathbf Z\log |q|$. And this map is a morphism of groups.
In Berkovich's point of view on non-archimedean analytic geometry, the analytic space $A^{\rm an}$ associated with $A$ is a (locally contractible) topological space which is simply connected when $A$ has (potentially) good reduction, and whose universal cover is the (analytic space associated with) the multiplicative group when $A$ has multiplicative reduction (ie $|j(A)|>1$).
For abelian varieties of arbitrary dimension, say $g$, the picture is due to Raynaud. For abelian varieties with good reduction, one cannot say much, they are simply connected in Berkovich's sense.
At the opposite spectrum, there are the totally degenerate abelian varieties whose universal cover is a $g$-dimensional torus $\mathbf G_m^g$ for which tropical geometry is a powerful tool. Then, $A$ tropicalizes to a $g$-dimensional torus. This picture has been very important for recent work in diophantine geometry, such as Gubler's proof of the Bogomolov conjecture abelian varieties over function fields which are totally degenerate at some place.
In general, the univeral cover of an abelian variety over a non-archimedean field is an extension of a simply connected abelian variety with good reduction by a torus of some dimension $t$. This gives a kind of partial tropicalization
to a real torus of dimension $t$.
