I don't think it's natural to say those in English.
In Filipino, we don't have a generic way to express that. In the case of eel-catcher, we could use the prefix "taga~" to say something is someone's job or responsibility, thus, eel-catcher is "taga-huli ng palos." However, if you want to say eel-eater just to tell someone that you like eating eel or that you prefer eating it over tuna, you can't use "taga~" or any other special affixations or forms. We just say them the long way. For instance, we would say "Mahilig akong kumain/umorder ng palos." To put more emphasis on eel you could say "Palos ang hilig kong orderin."
Yes, even an English linguist who is English himself admits that that is a rare, sloppy kind of wordings.
In Japanese, it goes like: "君 は 何 に する？ (What are you going to have/order?) ” － "ぼく は ウナギ だ。”
It's a very very famous sentence among Japanese linguists because its semantics has been puzzling them for decades.
In that case, I am very much familiar with that sentence construction. In fact, I use it when speaking Japanese. We do the same in Tagalog, though not as often as in Japanese.
For example: Ano'ng oorderin mo? Ako pasta.
It does sound like one is saying "I'm pasta," but it is actually short for "Ako, pasta ang oorderin ko." To avoid confusion though, others would prefer saying "(Sa) akin, pasta" which is literally "For me, pasta."