Shor's algorithm and Grovers algorithm are fundamentally very different. Most conventional asymmetric crypto algorithms are essentially all hidden sub group problems which are NP hard for classical computers. What Shor's algorithm does is reduce hidden sub group problems to P on quantum computers.
Grover's algorithm is quite complex and I'm not qualified to say much about how it works, but I do know that the underlying mathematics is very different.
Shor's algorithm absolutely does not reduce NP hard problems to P (or BQP). These kinds of problems with quantum speedups reside in an intermediate class of difficulty sometimes called NP-intermediate which may or may not already be in P, and which NP complete problems do not reduce to.
But it's also about another of Peter Shor's famous results: the Threshold theorem! But only in the sense that the authors seem to have never heard of it. (See my other comment.)
Grover's algorithm is quite complex and I'm not qualified to say much about how it works, but I do know that the underlying mathematics is very different.