I assume you also want your compact sets to have non-empty interior, hence positive volume.

The literature mostly deals with the related Banach-Mazur metric $d_{BM}(A,B)$, in which it is assumed that $\alpha_1(A)$ and $\alpha_2(B)$ are homothetic, so $d_{BM}(A,B) \ge d(A,B)$. (Here I'm following your convention and making $d_{BM}$ a metric, as opposed to the usual definition which makes its logarithm a metric.) Here's a little of what's known about that related to your question:

If $B$ is a Euclidean ball, then $d_{BM}(A,B) \le \log n$, with equality achieved exactly when $A$ is a simplex. Thus the diameter of $(\mathcal{C}_n, d_{BM})$ is at most $2\log n$. I believe the exact diameter is an open question.

Let $\mathcal{C}_n^0$ be the set of affine equivalence classes of centrally symmetric convex bodies. Then if $B$ is a Euclidean ball, $d_{BM}(A,B) \le \log \sqrt{n}$, with equality achieved when $A$ is a cube or a crosspolytope (but not only then); therefore the diameter of $\mathcal{C}_n^0,d_{BM})$ is at most $2\log\sqrt{n} = \log n$. Gluskin proved that the diameter of $(\mathcal{C}_n^0,d_{BM})$ is at least $\log \sqrt{n} + c$ for a constant $c$ independent of $n$, by in fact proving the same lower bound for the diameter of $(\mathcal{C}_n^0,d)$.

I assume you also want your compact sets to have non-empty interior, hence positive volume.

The literature mostly deals with the related Banach-Mazur metric $d_{BM}(A,B)$, in which it is assumed that $\alpha_1(A)$ and $\alpha_2(A)$ are homothetic, so $d_{BM}(A,B) \ge d(A,B)$. (Here I'm following your convention and making $d_{BM}$ a metric, as opposed to the usual definition which makes its logarithm a metric.) Here's a little of what's known about that related to your question:

If $B$ is a Euclidean ball, then $d_{BM}(A,B) \le \log n$, with equality achieved exactly when $A$ is a simplex. Thus the diameter of $(\mathcal{C}_n, d_{BM})$ is at most $2\log n$. I believe the exact diameter is an open question.

Let $\mathcal{C}_n^0$ be the set of affine equivalence classes of centrally symmetric convex bodies. Then if $B$ is a Euclidean ball, $d_{BM}(A,B) \le \log \sqrt{n}$, with equality achieved when $A$ is a cube or a crosspolytope (but not only then); therefore the diameter of $\mathcal{C}_n^0,d_{BM})$ is at most $2\log\sqrt{n} = \log n$. Gluskin proved that the diameter of $(\mathcal{C}_n^0,d_{BM})$ is at least $\log n - c$ for a constant $c$ independent of $n$, by in fact proving the same lower bound for the diameter of $(\mathcal{C}_n^0,d)$.