You can't pick your family...
The Shiba clan were one of the great noble families in Soul Society. However, they fell from grace some time ago, and have since fallen lower and lower. The two remaining Shibas, Kuukaku and Ganju, no longer even live in Soul Society. Instead they live out in the Rukongai, where normal spirits dwell. Both have high spiritual power, however. The event which caused the family's downfall is unknown, but it seems to have taken place a significant period of time ago.
So what is the connection to Isshin? First off, all the Shibas are highly eccentric. Isshin would fit right in. Secondly, Ichigo greatly resembles Kaien, the now deceased brother of Kuukaku and Ganju and former vice-captain of the thirteenth division. Both Rukia and Byakuya notice this similarity immediately, and this is possibly what causes Rukia to trust Ichigo so quickly. Kaien was killed by Rukia after a Hollow merged with his body, with no chance of Kaien returning, leading to much angst. His death greatly affected all who knew him; Kaien was a good man who cared about his subordinates.
Recent developments have proved that it is highly unlikely that Ichigo is the reincarnation of Kaien. Anyway, reincarnation has never been mentioned and it just seemed so unlikely. I also doubt that Ichigo so resembles Kaien just by coincidence, and I'll be disappointed if this is so.
The remaining conclusion is that they are related, which brings us right back around to Isshin. Namely, his departure from Soul Society and the Shiba clan's fall from nobility. What are the chances that these events are related? Its clear that Isshin wasn't using the name Kurosaki when he was a Captain. And the Shiba clan were powerful in the past, and most likely well known (if the Kuchiki clan are anything to go by). Indeed, it is upon hearing Ganju's name that Byakuya decides to actually pay him attention, even if he obviously has no amiable feelings towards him. Whatever caused the Shiba name to be stricken from the noble records must have been huge. We don't know whether Isshin left at the same time as Urahara and Yoruichi, but the Shihouin family have not had the same misfortunes as the Shiba. And if Isshin did leave Soul Society to be with Masaki, then she could not have been human herself, as his departure took place more a lot longer than twenty years ago.
Really, we have nothing to go on. If Isshin was part of the Shiba clan, he was probably head of the family, what with him being an obnoxiously powerful Shinigami captain. Whatever he did, be it leaving with another shinigami (Masaki) to live in the mortal world, or helping Urahara with forbidden experiments, it caused his family to suffer the consequences. Or perhaps he had no choice, and was an early casualty of Aizen's cause. Until Isshin decides to take a more active role in the conflict, we won't know.
Or maybe Kaien and Ichigo's resemblance is just coincidence.