Mid 3.0, but now I'm just being pedantic.

Keep it together; you're going to be senpai for green tanks in a few days!

Err. Abridged ver.: yes and no.

More complex answer: unlike Midgardsormr, Nidhogg didn't leave a corpse behind. Instead, the only thing that remained of him was his eyes, which yes, are stated to be the source of a dragon's life essence and Nidhogg's specifically to contain his rancor. The difference between what Midgardsormr and Nidhogg did was that Middy used the aether from sealing our Blessing to reconstruct his body without a host, while Nidhogg's rancor (hate, rage, etc. for people who don't follow the baroque language of the English version) possessed Estinien and reconstructed his body.

Of course, since Nidhogg was little more than hate and rage by the end, it makes little difference, but aside from Midgardsormr's initial comment he is referred to only as Nidhogg's "rancor" and "shade." Functionally there's no distinction, but technically Estinien-Nidhogg is different from the genuine article. (I think the Japanese version even calls him a "demon dragon" as opposed to a celestial dragon or somesuch.)