There are a couple of issues:

The first thing is that **isomorphism classes don't form a category**.

You may force them into being a category by picking a representative in each isomophrpism class, and then looking at the full subcategory that they span.
But this construction **depends** on the choice of representative:

If you change your mind, and pick another set of representative, then there is no **canonical** equivalence of categories between those two full subcategories (there are many equivalences of categories between those two categories, but there is no canonical one).

Ok, maybe that's not too bad after all...
This thing that you call "the category of equivalence classes" actually has a name: it's called a skeleton of the category.

Ok, now let's look at another, more serious isue:

A monoidal category (C,⊗) isn't just a category with a ⊗, it's also equipped with an associator α:(X⊗Y)⊗Z→X⊗(Y⊗Z).

If you restrict to a skeleton T⊂C of the category, then in order to equip that skeleton with a monoidal structure, you should pick for every object X of C an isomorphism
$$
\beta_X:X\to \underline{X},
$$
where $\underline{X}$ is the chosen representative on the equivalence class $[X]$.

The monoidal product on $T$ is given by
$$
X \underline\otimes Y := \underline {X \otimes Y}
$$
and it has the following associator:

$$ (X \underline\otimes Y) \underline\otimes Z = \underline{X \otimes Y \otimes Z} = \underline{\underline{X \otimes Y} \otimes Z}
$$

$$\xrightarrow{\beta^{-1} _ {\underline{X \otimes Y}\otimes Z}}\underline{X \otimes Y} \otimes Z \xrightarrow{\beta^{-1}_{\underline{X \otimes Y}}\otimes 1}(X \otimes Y) \otimes Z $$

$$\xrightarrow{\alpha} X\otimes (Y \otimes Z) \xrightarrow{1\otimes \beta_{Y \otimes Z}} X \otimes \underline{Y \otimes Z} $$

$$\xrightarrow{\beta_{X \otimes \underline{Y \otimes Z}}} \underline{X \otimes \underline{Y \otimes Z}} = \underline{X \otimes Y \otimes Z} = X \underline\otimes (Y \underline\otimes Z) $$

This associator goes from an object to itself, but is probably not trivial.
So the resulting monoidal category is maybe skeletal, but certainly not strict.