I don't know exactly what their motivations are, or even if they're concerned with Dollar Shave Club infringing on their patent. But, what I do know is that the patent that Gillette mentions seems to be a real thing that has been around for a while, they have the money and influence to make something happen so I wouldn't be surprised if they did.
Patents should be protected. Patents provide an incentive for companies to invest in research and development. This in turn adds more knowledge and technological advancements to society. Without patent protection, firms don't have a financial incentive to innovate, and ultimately, society loses out. It's highly likely Gillette has invested into it's razor technology, and even more likely that Dollar Shave might have accidentally copied the innovations which are probably now common place.
Yes, the claim is valid because Dollar Shave's razor blades were identical to those of Gillette's. If Gillette filed a patent for these razor blades and it was violated, then that all the more makes this infringement claim valid. Gillette's business suffered from this as a result, so Dollar Shave shouldn't continue to use the same razor blades.
In the world of patent infringement, Dollar Shave is now in litigation against one of the men's major grooming business concerns. This case is interesting . Can the legality of the case against Dollar Shave hold up under scrutiny by legal minds? As far as I can determine, this case is not valid since the area of infringement under question here is not a clearly defined issue.