Is it possible that Isshin, is related to the Shiba clan, as close as siblings to Kukkaka, Ganjyuu and Kaien? Why was the Shiba clan disowned? What if it had something to do with the eldest sibling (Isshin) abandoning his position of Captain and leaving SS with Urahara and Yoruichi? Now, reason for why the name Kurosaki is not known in SS, in Japanese tradition children keep the name of their mother, not their father. Maybe Isshin just took her name too so as not to make the link too obvious and putting his children in danger from old grudges from SS.
If you think about it, the whole family is hot headed. I mean, Isshin isn't as hot headed as Ichigo, but Urahara did once mention that he was very similar to his son. And what is Ichigo famous for? His hot headed-ness. Kukkaku? Being a hot headed female of Bleach. Ganjyuu? Just as much as his sister. Kaein's a happy go lucky guy, but I somehow sense a hot headed vibe from him too. The biggest factor is Ichigo's dad. Isshin is a captain level shinigami and what's more no real head of the Shiba clan who looks like they should be has been introduced. Kukkaku is obviously the current leader of the Shiba clan, but she is in same generation as Ganju and Kaien (since they've always been refered as siblings). Biggest possibility is that Isshin could have been the head of the household as the eldest brother of the other 3 Shibas, due to all of the resemblances.
The Kaien-ichigo relation though is actually mentioned in the manga between Ukitake and Byakuya after Ichigo drops down in an attempt to save Rukia, Ganju and Hanataro. I just think Byakuya was mistaken about there being no relation. Too many similarities lean toward it. Considering Isshin and Urahara are best friends, much like Ukitake and Kyoraku.