A Riemannian manifold $M$ is flat if it has constant curvature zero. It is called isotropic if all directions are equal under symmetry, that is for $p\in M$ and unit vectors $v,w\in T_p M$ there is a $\phi\in\mathrm{Aut}(M)$ with $\phi(p)=p$ and $\phi_*(v)=w$.
Question: is there a compact flat isotropic manifold of dimension $d\ge 2$?
I suspect that it is safe to say that this is not possible in dimension $d=2$, so I care most about $d=3$.
Compactness is neccessary as otherwise we can choose $\Bbb R^d$.
First, your definition is not quite right, isotropy requires $v, w$ to be unit tangent vectors.
There are no such manifolds in any dimension $n\ge 2$. If such a manifold $M$ were to exist, it would be the quotient of the flat ${\mathbb R}^n$ by a discrete (and torsion-free) group $\Gamma$ of isometries. Lift the isometry group of $M$ to the universal covering; the result is a subgroup $G$ of Euclidean isometries normalizing $\Gamma$. By the isotropy assumption, for each (enough, some) $q\in {\mathbb R}^n$ its stabilizer $G_q$ would act transitively on the unit tangent sphere. It is a nice exercise to conclude from this that the group $\Gamma$ then would be non-discrete (this is easier if you already know Bieberbach's theorems, which imply that $\Gamma$ contains a nontrivial translation). A contradiction. qed