If $G$ is a unimodular second countable Type I group, then the Plancherel measure is the unique measure $\mu$ such that
$$\|f\|_2^2 = \int_{\widehat{G}} \|\pi(f)\|_{\mathrm{HS}}^2 \mathrm{d}\mu(\pi).$$
for every $f \in \mathrm{L}^1(G) \cap \mathrm{L}^2(G)$. This appears as Theorem 18.8.2 in Dixmier's book on $C^*$-algebras.
When $G$ is not unimodular, the question becomes more complicated, because the Plancherel measure needs to be twisted by a section of a line bundle; see the paper of Duflo-Moore on the subject for the gory details. When $G$ is not second countable, I do not know of a published result; the technical details of direct integral theory are more difficult in this case and not standard. When $G$ is not Type I, the decomposition of the left regular representation into irreducibles is no longer unique, and some of the operators on the right-hand side of the formula will fail to have finite Hilbert-Schmidt norm.
Edit: The question has been clarified to indicate that it is not asking for this characterization of the Plancherel measure, but rather a characterization that is closer to the description in the abelian case of the Plancherel measure as a Haar measure on $\widehat{G}$. Since there is usually no algebraic structure on $\widehat{G}$ for which to consider invariance, a direct generalization is impossible.
The closest analogue is the characterization of the Plancherel measure as a unique co-invariant trace (or weight) on the von Neumann algebra $\mathcal{M}$ generated by the left-regular representation of $G$. Suppose $G$ satisfies the same hypotheses as above and $\Delta : \mathcal{M} \to \mathcal{M} \overline{\otimes} \mathcal{M}$ is the comultiplication on $\mathcal{M}$ given by $\lambda(s) \mapsto \lambda(s) \otimes \lambda(s)$. Then the Plancherel trace is the unique normal semifinite trace $\tau$ on $\mathcal{M}$ such that
$$\tau((\varphi \otimes \mathrm{id}) (\Delta(a))) = \tau(a)$$
for all $a \in \mathcal{M}_\tau^+$ and $\varphi \in \mathcal{M}_*$. A similar characterization holds for the Plancherel weight of an arbitrary locally compact group, or for the Haar weight of a locally compact quantum group. For proofs, see volume 2 of Takesaki or any of the literature on von Neumann algebraic quantum groups.
