We can assume that these two characters are not the same person since the games have a huge time gap between them, although it would seem that Eri Karst and Eri in the fourth game are indeed related in some way since both are Kartian Princesses.
Eri Karst is a secondary character and boss. She's fought twice during the game.
Eri is the princess of Karst. She disguised herself and fled to Koho in hopes of finding a "miraculous power" said to be housed inside a shrine and use it to save her people from miserable living conditions.
Due to her prideful and aggressive personality she clashes twice against Momo Reinol without giving an explanation. After being defeated a second time she confesses her identity and motivations to Momo, thanks her for putting some sense into her, and admits she didn't have any knowledge of the existence of an Underworld Queen inhabiting the shrine.
She leaves the area after this, presumably back to her homeland.
Momodora: Reverie Under the Moonlight
Eri is a minor character with which the player interects several times throughout the story.
She's initially found inside the Subterranean Grave, having escaped imprisonment. At first she's startled by Kaho Reinol's sudden appearance, before lowering her guard and advising the priestess to run away from Karst City.
She's later found again in Karst City. When talked to she will express a complete fatalism at the future, even considering her escape just a temporary solution at best. Depending on Kaho's answer she may remain that way or become more optimistic. The answer itself doesn't have particular effect aside from changing dialogue.
She's found one last time in the Sacred Ordalia Grove, near the gates to Karst City, where she's taking refuge from the curse. Her dialogue will vary depending on the answer given during the previous encounter. At this point she may offer to trade her Blessing Charm for the Hazel Badge. She remains here for the rest of the game.
- It's heavily implied Momodora: Reverie Under the Moonlight's Eri is the royal princess of Karst, which would make her the Accurst Queen of Karst's relative. Besides sharing her name with Momodora II's Eri, the Blessing Charm's description mentions it's only passed down infants of the Karstian Royal Family.