$$∀x∀y ((\text{Cube}(x) ∧ \text{Cube}(y)) → x = y)$$
To me, this statement means that for all cubes x and y, if you pick up a cube from the box, it will always be the same cube.
I understand this to mean that there is exactly one cube in the box.
I don't understand how the statement means that there is at most one cube in a box.
$$∀x∀y\;\big((\text{Cube}(x) ∧ \text{Cube}(y)) → x = y\big)$$
Yes the above statement allows for the possibility of exactly one cube in the box, as you have explained.
But it also allows for the possibility of no cube in the box: in this case, whichever pair of objects you pick, $(\text{Cube}(x) ∧ \text{Cube}(y))$ is false and the entire statement consequently vacuously true.
Here's an equivalent, more succincct formulation of at most one: $$\exists x\forall p\;(\text{Cube}(p)\to p=x).$$
In contrast, exactly one cube can be translated as $$\exists x\forall p\;(\text{Cube}(p)\leftrightarrow p=x).$$