Can Floyd-Warshall be used to solve an APSP problem without copying the matrix?
No you are not lucky, it is possible to use the program that way. The notation $d_{ij}^k$ probably is used for better explaining what the algorithm has achieved in the $k$th step. Note that the variant where one overwrites the matrix as you suggest is presented in Exercise 25.2-4 (in my second edition of the book, which is not the latest).

The secret to that puzzle is in principle that an assignment to $d_{i,j}^k$ is only done once, and that value is never again used in the computation, except for the case where either $i=k$ or $j=k$. However, the algorithm does not change those values in the $k$th step. You figure out why.