Those do look more and more like some sort of abrasions, that have resulted in missing scales. I don't think they are bites from anything, nor parasites. Is your substrate very sharp-edged and abrasive? Sharp-edged rocks or driftwood in the tank? Do those drainpipe sections perhaps have some sharp edges either inside or at the openings?
Fire Eels are often referred to as "scaleless" but they are fully scaled with numerous small scales, as can easily be seen in your pics. A big fish squeezing into and around anything abrasive can easily sustain this kind of damage.
I had the larger gutter pieces for a while and it didn't have these abrasions. Unless it's from the thinner gutter piece and it's still healing.
Would you happen to have any recommendations for hide out spots for the eel? Gutters didn't appear to have sharp edges but I'll have to double check.
Makes me feel better that it looks like abrasions.