Well, there are also legless salamanders, that look like eels pretty much.

https://en.wikipedia.org/wiki/Two-toed_amphiuma

Some of them have no lungs even:

https://en.wikipedia.org/wiki/Microcaecilia_iwokramae

So, it makes sense to say that eels are fish, because there are lungless eel-like creatures that are actually amphibians.