Could the Enigma algorithm be classified as a Feistel network?
Enigma is not a Feistel cipher. A "Feistel cipher" is a block cipher with a specific structure, namely the whole business with the two halves, the combination of one half with a (one-way) function of the other half and a reversible operation (e.g. XOR), and the swap. See the Wikipedia page which has nice schematics.

So considering Enigma as a kind of Feistel cipher is like considering a salmon as a kind of chicken: yeah, it has no beak, no feathers, no legs and no wings, but could it be a called a chicken ? Strangely enough, I tend to answer no to that. Even though both the salmon and the chicken are edible. But being edible is not sufficient to grant chickenness. Similarly, being a symmetric cipher is not sufficient to grant Feistelness.