Since I don't have a good understanding of your Classes axiom/theory, let me answer instead for Gödel-Bernays set theory GBC, for which the answer is negative.

We know that there are pointwise definable models of GBC, and every such model will satisfy your definability axiom, since every class there is definable. But none of these models think that every set or class is countable.

The main explanation is that merely knowing that every class is definable is not sufficient to build the definability map $$\text{class }X\quad\to\quad\text{defining formula }\phi.$$ It is this map and not the pointwise definability itself that leads to the conclusion that there are only countably many classes.

We discuss this issue at length in our paper:

• Hamkins, Joel David; Linetsky, David; Reitz, Jonas, Pointwise definable models of set theory, J. Symb. Log. 78, No. 1, 139-156 (2013). ZBL1270.03101.

But I am unsure how much of this analysis applies to your theory Classes.

Since I don't have a good understanding of your Classes axiom/theory, let me answer instead for Gödel-Bernays set theory GBC, for which the answer is negative.

We know that there are pointwise definable models of GBC, and every such model will satisfy your definability axiom, since every class there is definable. But none of these models think that every set or class is countable.

The main explanation is that merely knowing that every class is definable is not sufficient to build the definability map $$\text{class }X\quad\to\quad\text{defining formula }\phi.$$ It is this map and not the pointwise definability itself that leads to the conclusion that there are only countably many classes.

We discuss this issue at length in our paper:

• Hamkins, Joel David; Linetsky, David; Reitz, Jonas, Pointwise definable models of set theory, J. Symb. Log. 78, No. 1, 139-156 (2013). ZBL1270.03101, MR3087066.

But I am unsure how much of this analysis applies to your theory Classes.