It seems clear that every abelian surface $X$ can be realized as an anti-canonical divisor since we can view $X$ as an etale double cover of another abelian surface $Y$ and so $X$ embeds as an anti-canonical divisor in the threefold $\mathbb{P}(\mathcal{O}_{Y}\oplus L)$ where $L \to Y$ is the two torsion line bundle corresponding to the cover $X \to Y$. So the OP seems to be asking - which smooth K3 surfaces can be realized as anti-canonical divisors of smooth threefolds. I am not sure where this is going since obviously every K3 is a connected component of an anti-canonical divisor.

In general the two projective bundles are related by a flip and accidentally the result of the flip can be isomorphic to the original bundle. The point is that the natural rational map given by the elementary modification is never and isomorphism and is resolved explicitly by a flip. As a bit of shameless self-promotion - take a look at Apppendix A of arxiv.org/abs/math/0008010 where this is explained and where you can find the references to the classical papers of Tyurin and Maruyama who analyzed this in detail.