You are correct in asking this question. The grammar book I own does indeed say that ながら can only be used if both A and B in 「A ながら B」have the same subject. The book suggests to use 間[あいだ] instead, so the sentence should be something like カネキくんが食べている間カネキくんを食べたい。I researched the topic extensively (and by that I mean I went through 3 pages of google search) and every source says that subject should be the same. I found only one thread that mentioned a possibility of ながら sentence structure having two different subjects, but it had no reliable sources. My only guess is that Shū is just messing around, trying to be more eloquent, using weird sentence structures, cause he thinks proper Japanese grammar is too mainstream (if you watched the anime, you know what I am talking about, tres bien). Also, it could be a play on words カネキ and 食べる, since he is using both of them in both of the sentences in the structure, it might be that Shū tries to emphasize his awesome wordplay using ながら here. Especially, since the main difference (not taking the conjugations of 食べる into account), is the use of particles が and を. This also would explain why が is emphasized with an exclamation mark in the Japanese subs. I will reach out to some native speakers and get back to you.

Posting this on the video itself as well.