This is refutable in ZFC for singular cardinals $\kappa$, since by König's theorem $\kappa<\kappa^{\text{cof}(\kappa)}$.
So for some sets $A$, such as $A=\aleph_\omega$, the set $B$ is definitely larger than $A$.
For other sets, $A$, it can be true. Set theorists would usually write $\kappa^{<\kappa}=\kappa$ for the cardinals $\kappa$ for which it holds, and this is commonly seen as a hypothesis in theorems. If GCH holds, then this is true for every regular cardinal (and false for every singular cardinal).