I will have to look for a reference, but strong Morita equivalence is the notion of equivalence in the $\rm C^*$-2-category of $\rm C^*$-algebras, Hilbert $\rm C^*$-correspondences, and adjointable intertwiners.
In more detail, there is a weak 2-category (a.k.a. a bicategory) $\rm C^*$-Alg whose objects are $\rm C^*$-algebras (let's assume they are unital for ease of exposition) and whose 1-morphisms are right Hilbert $\rm C^*$-correspondences. This means a 1-morphism $X: A\to B$ is a Banach space equipped with a right $B$-valued inner product such that the norm on $X$ is equal to $\|\langle x|x\rangle_B\|_B^{1/2}$, together with a left $A$-action by adjointable operators. The 2-morphisms $X\Rightarrow Y$ are the adjointable $A-B$ bimodular operators. 1-Composition is the relative tensor product of correspondences. [I'm sure there is a better reference for all this, but I'm pulling all this from my joint article arXiv:2105.12010 since I know it best. It also includes many details on $\rm C^*$-2-categorie with the $\rm C^*$-algebra community specifically in mind.]
Now in a 2-category, an equivalence between 2 objects $A,B$ is a pair of 1-morphisms $X: A\to B$ and $Y: B\to A$ together with two 2-isomorphisms ${}_AX\otimes_B Y_A \Rightarrow 1_A$ and ${}_BY\otimes_A X_B\Rightarrow 1_B$. Here I am writing 1-composition left-to-right, suggestively using relative tensor product notation. When the 2-category is a $\rm C^*$-2-category, one further requires that these two 2-isomorphisms are unitary. One then checks that two (unital) $\rm C^*$-algebras are strongly Morita equivalent if and only if they are equivalent in the 2-category $\rm C^*$-Alg. It is straightforward to prove that an imprimitivity bimodule gives an equivalence in $\rm C^*$-alg, but the other direction is a bit more subtle, and requires me to discuss adjoints, unitary adjoints, and adjoint equivalences.
First, in a 2-category, there is a notion of left and right adjoint of a 1-morphism ${}_AX_B$, similar to the notion of adjoint functor. (Indeed, the notion of adjoint functor is exactly the notion of adjoint in the 2-category Cat.) We call a 1-morphism adjointable if it admits both a left and a right adjoint. The composite of two adjointable 1-morphisms is again adjointable, so we may restrict to a 2-subcategory whose 1-morphisms are adjointable. This 2-subcategory necessarily contains the equivalences, and it is a fact that every equivalence in a 2-category can be promoted to an adjoint equivalence, i.e., an equivalence which satisfies the extra zig-zag/snake equations.
In a $\rm C^*$-2-category, however, one should be careful to choose adjoints of 1-morphisms which are compatible with adjoints of 2-morphisms. It is at this stage that we are calling too many things "adjoints," which can lead to confusion. So in the rest of this answer, I'll use the term "dual" for an adjoint of a 1-morphism and use the term "adjoint" only for linear operators. The notion we are looking for here is that of unitary dual functor from arXiv:1808.00323. Typically, there are many that one can put on a given $\rm C^*$-2-category, and this article explores this in detail. In contrast, in the non-$\rm C^*$-setting, there is a contractible choice of dual functor; you can read the introduction to that paper if you're interested in that story.
Now, it is well-known by [MR2085108 = arXiv:math/0301259, MR1624182] that a Hilbert $\rm C^*$-correspondence is dualizable if and only if it is finitely generated projective (fgp) on both sides. Restricting to the fpg 1-morphisms in $\rm C^*$-Alg, it should be the case that there is a unitary dual functor where the dual is given by the contragredient bimodule and the evaluation map $\operatorname{ev}_X: {}_B\overline{X}\otimes_A X_B\to {}_BB_B$ is given by the $B$-valued inner product. At this step, I realize there is really something to check, as ${}_AX_B$ was only equipped with a right $B$-valued inner product, and not a left $A$-valued inner product. If my memory serves me correctly, I think we can equip a dualizable 1-morphisms ${}_AX_B$ with such a left $A$-valued inner product so that everything here works out; the references here are almost certainly [MR2085108 = arXiv:math/0301259, MR1624182].
Now once we have equipped the dualizable 2-subcategory of $\rm C^*$-Alg with this unitary dual functor, we see that an adjoint equivalence here should be an imprimitivity bimodule giving a strong Morita equivalence in the sense of Rieffel. There are some steps to check here, but I think it should all work out. The last thing to point out here is that in the $\rm C^*$-setting with a fixed unitary dual functor, it is still true that every equivalence can be promoted to an adjoint equivalence.
As you mention in the question, this gives a categorical proof that the categories of Hilbert space representations are equivalent. Observe that given a (unital) $\rm C^*$-algebra $A$, a Hilbert $\rm C^*$ $A-\mathbb{C}$ correspondence $H$ is indistinguishable from a Hilbert space $H$ equipped with a left $A$-action. Thus the $\rm C^*$ hom 1-category $\operatorname{Hom}(A\to \mathbb{C})$ is the $\rm W^*$-category $\operatorname{Rep}(A)$ of Hilbert space representations of $A$. Given an equivalence 1-morphism ${}_AX_B$, pre-composition with $X$ gives an equivalence of categories ${}_AX\otimes_B - : \operatorname{Rep}(B) \to \operatorname{Rep}(A)$.
However, we can also look at homs the other way. That is, the $\rm C^*$ hom 1-category $\operatorname{Hom}(\mathbb{C}\to A)$ is the $\rm C^*$ category $\operatorname{Mod}(A)$ of right Hilbert $\rm C^*$-modules. Again, given an equivalence 1-morphism ${}_AX_B$, post-composition with $X$ gives an equivalence of categories $-\otimes_AX_B: \operatorname{Mod}(A) \to \operatorname{Mod}(B)$.
While an equivalence $\operatorname{Rep}(A) \cong \operatorname{Rep}(B)$ is too weak to capture strong Morita equivalence, I suspect the equivalence $\operatorname{Mod}(A) \cong \operatorname{Mod}(B)$ fully captures the notion.
