The parallel has been made before, and yeah, I think there are some parallels with Rasputin – her evangelical piety, prophetic tendencies, her mesmeric force of personality, and her intense sexual charisma. However, there’s one big missing element: there’s no evidence of Melisandre ever promising to cure Shireen, because if she had there would be some explaining to do about why she hasn’t succeeded. So Selyse’s faith came from elsewhere else.
Indeed, in the books, Selyse’s attraction to the Faith seems attached to the idea that Stannis is Azor Ahai reborn and R’hllor will make him king, and the more forward-looking idea from Davos V of ASOS that:
“Your own wife begs as well, lord husband.” Queen Selyse went down on both knees before the king, hands clasped as if in prayer. “Robert and Delena defiled our bed and laid a curse upon our union. This boy is the foul fruit of their fornications. Lift his shadow from my womb and I will bear you many trueborn sons, I know it.” She threw her arms around his legs. “He is only one boy, born of your brother’s lust and my cousin’s shame.”
So rather than looking for a cure for Shireen, Selyse seems attracted instead to the idea that R’hllor can restore her fertility and allow her to bear a male heir.