They just end with the same sound, somewhat like how "carbon" and "carpet" begin with the same sound but they do not at all share the same Latin roots despite both coming from Latin if you look back far enough. Due to the syllabic nature of Japanese, "etsu" (エツ) is not a part of "setsu" (セツ) or "getsu" (ゲツ), as odd as it may seem when you spell it alphabetically. (using kana here, since that's what the NPCs in game are spelt with; it's an often-exercised convention in fiction, but I fear explaining it fully might be more information than you wish)
Your question is perhaps better directed towards the lore forum, as all of those names are in fact words in modern Japanese.