The nlab has this paragraph on dagger categories (from: https://ncatlab.org/nlab/show/dagger+category) :
In enriched category theory, involutive categories have also been called symmetric categories. Sometimes the involution is required to be strict in the sense that the dagger is an equality rather than an isomorphism: in line with horizontal categorification, one could argue these should be called commutative categories, since a one-object category with such an involution is a commutative monoid.
The bolded claim looks false to me, because of the following counterexample:
If I consider 3x3 square matrices with transposition as my involution, the matrices aren't commutative because rotation isn't commutative. And these form a one object category, a monoid.
Where am I going wrong?
A one-object category $\mathcal C$ is exactly a monoid, in which case $\mathcal C^\circ$ is its opposite monoid. If we have an involution $\dagger \colon \mathcal C^\circ \to \mathcal C$, then every element $x$ of the monoid has an "adjoint" $x^\dagger$ such that $(x \circ y)^\dagger = y^\dagger \circ x^\dagger$. If $\dagger$ is the identity on morphisms, then we have $x^\dagger = x$, so that $x \circ y = y \circ x$, which is precisely commutativity.
In fact, this is the prototypical example of a dagger-category with a strict involution in this sense: as observed by MPos in the comments, for every morphism $f \colon A \to B$ in a dagger-category with a strict involution, we must have $f^\dagger = f$, which implies that $A = B$. Thus every dagger category with a strict involution is a coproduct of commutative monoids. Thus, the notion is not particularly interesting. However, it gives some intuition for seeing dagger-categories as a categorification of commutative monoids.
Regarding your example, it is not the case that, for every matrix $M$, we have $M^\top = M$: this is only true when $M$ is symmetric. Therefore your example is not a counterexample: it is an example of a dagger-category where the involution is not the identity on morphisms.