I don't know what this is all about. A eulogy is indeed a funeral oration, but an elegy is also an act of mourning, a poetic one (from Greek elegos, mournful song). The American Heritage Dictionary says this:
elegy:
- A poem composed in elegiac couplets.
2a. A poem or song composed especially as a lament for a deceased person. b. Something resembling such a poem or song.
- (Music) A composition that is melancholy or pensive in tone.
In general, a eulogy is also an elegy (definition 2b).