It looks like shinigami powers are not passed on like flesh and blood. I have a theory that divides in two scenarios. Quincy powers have been confirmed to be passed from the parents, and it has been confirmed, twice in a month, that Ichigo is not a Shinigami. He is a quincy and a Fullbringer. What if Isshin didn't lose his shinigami powers because of Urahara's gigai but as a result of having them absorbed by being near Ichigo and his sisters? Then Ichigo suddenly manifested those Shinigami powers he had been absorbing with his uncontrolled quincy powers for so long and combined them with the hollow reishi passed on from his mother (and in essence with his fullbring) and he either fullbrought those powers into a fake shinigami state or with something we still don't know. When Ichigo finally left his home to train with Urahara, went to SS and then to Hueco Mundo, Isshin had time to replentish his powers and became a Shinigami again. Maybe that's how he got a Zanpakuto with similar abilities and a similar name.
Now you might think it is because they are family but if you remember Shiba Kaien's zanpakuto was NOTHING like Zangetsu/Engetsu, at all!
It was a chinese halberd that created and controlled water!
Or Isshin might as well have passed on his powers to Kugo Ginjou; Kubo just said in Germany that Kugo was necessary because Rukia needed to know of the process to create a substitute shinigami and who would tell her other than the nephew of the guy who did it?!
Lets just wait and see what Kubo came up with :3