A partial answer.
From the cohomological point of view, a compact Kähler manifold satisfies the "$dd^{\mathcal{J}}$-Lemma" with respect to any of its structures. In fact, such properties do not characterize Kählerness: for example, Fujiki class $\mathcal{C}$ manifolds and Moishezon manifolds satisfy $\partial\overline{\partial}$-Lemma, as Gunnar recalled.
More precisely:
with respect to the complex structure, a compact Kähler manifold satisfies the classical $\partial\overline{\partial}$-Lemma, stating that every $\partial$-closed $\overline{\partial}$-closed $d$-exact form is also $\partial\overline{\partial}$-exact, namely, the natural map $H^{\bullet,\bullet}_{BC}(X)\to H^{\bullet}_{dR}(X;\mathbb{C})$ induced by the identity is injective (here, $H^{\bullet,\bullet}_{BC}(X):=\frac{\ker\partial\cap\ker\overline{\partial}}{\mathrm{imm}\partial\overline{\partial}}$ is the Bott-Chern cohomology of $X$, and its dual $H^{\bullet,\bullet}_{A}(X):=\frac{\ker\partial\overline{\partial}}{\mathrm{imm}\partial+\mathrm{imm}\overline{\partial}}$ is the Aeppli cohomology). As a consequence, one gets the Hodge decomposition. In particular, $\dim_{\mathbb{C}} H^{\bullet}_{dR}(X;{\mathbb{C}}) = \sum_{p+q=k} \dim_{\mathbb{C}} H^{p,q}_{\overline{\partial}}(X)$ and $\dim_{\mathbb{C}} H^{p,q}_{\overline{\partial}}(X)=\dim_{\mathbb{C}} H^{q,p}_{\overline{\partial}}(X)$ for every $k$, for every $p,q$. Note that such equalities do not characterize even the validity of the $\partial\overline{\partial}$-Lemma. Furthermore, one has that a compact complex manifold satisfies the $\partial\overline{\partial}$-Lemma if and only if $2\dim_{\mathbb{C}}H^k_{dR}(X;\mathbb{C}) = \sum_{p+q=k} \left(\dim_{\mathbb{C}} H^{p,q}_{BC}(X) + \dim_{\mathbb{C}} H^{p,q}_{A}(X)\right)$ for every $k$.
with respect to the symplectic structure, a compact Kähler manifold satisfies the Hard Lefschetz Condition, which is the symplectic counterpart of the $\partial\overline{\partial}$-Lemma. By definition, Hard Lefschetz Condition states that $[\omega^k]\colon H^{n-k}_{dR}(X;\mathbb{R})\to H^{n+k}_{dR}(X;\mathbb{R})$ is an isomorphism for every $k$, where $\omega$ is the Kähler form and $2n$ denotes the real dimension of $X$. By considering the symplectic differential $d^{\Lambda}:=\left[d, -\iota_{\omega^{-1}}\right]$ and defining the Tseng and Yau symplectic Bott-Chern cohomologies as in the complex case by using $d$ and $d^\Lambda$ instead of $\partial$ and $\overline{\partial}$, one has that the Hard Lefschetz Condition is equivalent to the $dd^\Lambda$-Lemma, which is equivalent to the equality $\dim_{\mathbb{R}} H^k_{BC}(X) + \dim_{\mathbb{R}} H^k_{A}(X) = 2\dim_{\mathbb{R}} H^k_{dR}(X;\mathbb{R})$ for every $k$.
Looking at a Kähler manifold as a generalized-complex manifold (in the sense of Hitchin, Gualtieri, Cavalcanti,) one gets further numerical properties. For example, one gets that the $dd^{\mathcal{J}}$-Lemma with respect to the generalized-complex structure of type $n$ is equivalent to the equalities $\dim_{\mathbb{C}}H_{dR}^{k}(X;\mathbb{C})= \sum_{p+q=k}\dim_{\mathbb{C}}H^{p,q}_{\overline{\partial}}(X)$ and $\sum_{p-q=k}\dim_{\mathbb{C}}H^{p,q}_{BC}(X) = \sum_{p-q=k}\dim_{\mathbb{C}}H^{p,q}_{\overline{\partial}}(X)$ for every $k$ (note, "$p-q=k$" instead of "$p+q=k$").
Obviously, as said, these cohomological properties are quite far from being sufficient.. Notwithstanding, Kähler manifolds are characterized inside some special classes of compact manifolds, such as, for example, nilmanifolds and solvmanifolds.
Note also a related question: Fundamental Groups of compact Complex manifolds?