Let $X$ and $Y$ be two Fano varieties of the same dimension embedded into a same projective space $\mathbb P^N$, assume $Pic X= \mathbb Z\mathcal O\_X(1)$ and $Pic Y=\mathbb Z\mathcal O\_Y(1)$, where $\mathcal O\_X(1)$ means the restriction of $\mathcal O\_{\mathbb P^N}(1)$ on $X$. 


Then does it hold that $X$ and $Y$ are in fact isomorphic?


Such kinds of problems seem to have been fully understood due to its easy statement, but I do not have an idea on it.



Minimal embeddings of certain Fano varieties with Picard number one
This is definitely not true; just consider two smooth cubic threefolds in $\mathbb{P}^4$.


Of course many other examples exist, e.g. Fano hypersurfaces in projective space which are not quadrics nor cubic surfaces (here the Picard group is generated by the hyperplane class by the Lefschetz hyperplane section theorem). Also many classes of Fano threefolds of Picard number one.


If what you wanted was true, most of the interesting moduli spaces for Fano varieties would in fact be trivial.


