Does Morita theory hint higher modules for noncommutative ring?

Yes. The trick is to use not just categories, but pointed categories, which are categories equipped with a choice of object (the "pointing"). Given any ring $R$, the category $\mathrm{Mod}(R)$ is naturally pointed by the rank-1 free module, i.e. $R$-as-an-$R$-module, which I will write as $R_R$. Then it is almost trivial that the pointed category $(\mathrm{Mod}(R),R_R)$, up to equivalence, recovers $R$ up to isomorphism.

What's that? I didn't tell you what the morphisms are between pointed categories, so you don't what the equivalences are? Well, you do actually know what the equivalences are: an equivalence of pointed categories $(\mathcal{C},C) \simeq (\mathcal{D},D)$ is an equivalence of categories $F : \mathcal{C} \overset\sim\to \mathcal{D}$ together with an isomorphism $f : FC \cong D$. I mean, what else could it be? Anything else wouldn't justify the name. But there is actually an interesting question of what are the morphisms which are not equivalences. Surely, a morphism $(\mathcal{C},C) \to (\mathcal{D},D)$ should consist of a functor $F : \mathcal{C} \to \mathcal{D}$ together with a morphism $f$ between $FC$ and $D$. The interesting question is whether $f$ should be: (1) an isomorphism; (2) a morphism $f : FC \to D$; (3) a morphism $f : D \to FC$. These three options have names: (1) is called a strong pointed functor; (2) is called an oplax pointed functor; and (3) is called a lax pointed functor. It is almost trivial to show that all three options give the same notion of equivalence of pointed categories, but they give different bicategories of pointed categories (and this difference matters in applications).

An advantage of working with pointed categories is that there are plenty of pointed categories which share with $(\mathrm{Mod}(R), R_R)$ some of its nice structural properties, but not all, and so are not of that form.

Although I think my answer "pointed categories" is an important one, there is another way that the question could be interpreted: What is an interesting class of rings which are recoverable up to isomorphism from their categories of modules? And perhaps, if we are not trying to be too universal or functorial, we won't mind if the isomorphism is not unique.

For this, I strongly recommend Morita's original paper introducing his equivalences. (Kiiti Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83-142 (1958). You can find a PDF by googling.) Among the many things in that paper is a theorem saying that rings satisfying a natural minimality condition called "basic" are Morita equivalent iff they are isomorphic (commutative rings, I believe, satisfy this condition) and that every finite-dimensional ring is Morita equivalent to one satisfying this minimality condition. See my answer to a related question.