In the movie it says that the weapon X program "wakes up" mutant genes, and that either you mutate or die trying. This means that there are probably some people without these genes that die in the process of trying to mutate, and this suggests that Wade did have dormant mutant genes that were woken up after plenty of abuse.
Although these powers are very similar to Wolverines I do not believe there are any direct connection between them. Just coincidence.