There are no specific accounts of a particular sword named Ishikirimaru. The character seems to be influenced by various accounts.
The most obvious influence is the sword enshrined in Ishikiri-Tsurugiya Shrine in Osaka. The shrine itself has been in existence since 927 CE or earlier. It has a reputation for healing swellings and tumours, hence the references in the character's lines.
Another possible influence is the Kabuki play Kajiwara Heizo Homare no Ishikiri (The Stone-Cutting Feat of Kajiwara Heizo), which features a sword that could cut through stone.
A sword named Ishikiri also appears in the Heiji Monogatari. Said to be the work of Sanjou Munechika's son Arinari, it was the favoured sword of Minamoto no Yoshihira. This Yoshihira was widely feared for his fighting prowess and was thus nicknamed "Kamakura Akugenta" (The Daring Eldest Son of the Minamoto Clan of Kamakura).
Ishikirimaru repeatedly says that he is unused to battle after living in a shrine for centuries. However, when maxed out his Impact stat is the highest of all the swords. Perhaps this is the influence of Minamoto no Yoshihira's Ishikiri.
Interestingly, it is not clear if any of the swords in these accounts are actually Oodachi. Most likely, they are Tachi.
He has been nicknamed "Papa" by some fans, probably because of his personality and mannerisms.
As he has a Recollection with Nikkari Aoe, they tend to appear together in fanworks.