In order to master Shikai, a Shinigami has to learn his/her Zanpaktou's name.
In doing so, a Shinigami can master the Shikai, as we saw with Ichigo, before knowing his Zanpaktou's name, he could barely use it in his training with Urahara and his first real fight with Renji in the SS arc.

We then learn that Bankais has a name as well, but apparently this is the full name of the Zanpaktou.
In Ichigo's case it's Tensa Zangetsu, in Renji's case it's Souou Zabimaru.

As we know in Japanese, the family name is coming before the persons name, there seems to be some connection here?

So knowing the name in Bleach, really, really matters.

But then we learned that Ichigo did not know his "sources" and therefore wasn't able to utilize his full powers. Apparently Renji knew his source (Logically easier for him, he's only a Shinigami), and therefore passed the Asauchi test.

So Ichigo did not know his sources, but knew his Zanpaktou's full name.
Renji however knew his source, but did not know his Zanpaktou's full name.

Knowing your family name (Bankai part of the name) is the same as knowing your "source"? So essentially Ichigo and Renji had the same problem, but for different reasons.
Will the outcome be the same for the two of them? A significant boost in power?

One could take a lead of faith, and speculate if Isshin's Zanpaktou name would be Tensa Engetsu.
Perhaps, if Renji's family ever had become a Shinigami, his family/source heritage is the "Souou" part of the name.

Also a whole other ball game, is Ichigo's Quincy powers, does knowing the source, and name meaning anything for a Quincy, considering Quincy apparently is a polar opposite to Shinigami in powers?
What are your thoughts?