I believe Yamato was created to absorb energy. So, Eva's seal would activate Yamato's Angel Mode and Sparda's seal would activate Yamato's Demon Mode.
What's more important to ask is where did Dante get Ebony and Ivory from this time around?
He probably stole them considering DmC Dante's background here; DMC Dante and Nero had their signature handguns crafted since they were in fortunate situations - Dante wasn't ostracized and Nero was under The Order of the Sword's protection. Ninja Theory said the design for E&I in DmC was to reflect a sense of age, so maybe during a trip into Limbo, Dante was able to leave with these as "souvenirs". That or they're Dante's manifestation of power.
The Rebellion and the shield with the same symbol on Dante's back were shown with Sparda while Yamato and E&I were never mentioned to be related to Sparda at all. Vergil and Dante could have manifested Yamato and E&I as their respective physical powers. This kind of fits in with Vergil having more control of his demonic and/or angelic heritage than Dante; he was able to manifest a powerful and terrible sword while Dante was only able to manifest a pair of Desert Eagle-like pistols. It also fits in with Vergil having a more classy and sophisticated air by having a sword; back in the old days only people of status carried swords while regular grunts carried spears and other simple weapons. E&I would fit into Dante's more rebel," shoot first, ask questions later", and wannabe gangster look. Remember, this Dante doesn't have an established identity yet compared to DMC Dante. You could even say DmC is a coming of age story. After all, DmC Dante started off as a punk trying to be cool, tough guy who matures and becomes a selfless hero.
Anyway, Rebellion and the shield would then appear to take over Sparda's role as the twin's inheritance from their father. DmC's Rebellion being able to shape-shift and ability to tap into their father's (and mother's and other's) powers makes it much more similar to Sparda, the weapon.
The other thing is that Sparda possessed a neutral weapon, Rebellion. The Arbiter and Osiris are implied to be either weapons used by Sparda and Eva or just their powers given form; the Arbiter is a crushing force while Osiris is a graceful and beautiful weapon. So, the other wild theory could be that Yamato belonged to Sparda similar to the history of DMC. In DmC, before leaving Vergil, Sparda may have given Yamato to him. The graffiti during Dante and Vergil's talke has Yamato looking wicked and demonic while Rebellion remains as is. So, Yamato may have more demonic origins: Sparda. E&I are still a mystery. Maybe they're Eva's gifts to Dante. Kind of like having two guardian angels for her son. That and in DMC, E&I had portraits of two women which could be referenced as "angels" in DmC that protect Dante.