flamo is from Latin flamma, via or influenced by many Romance and Germanic descendents, which Wiktionary gives the etymology "From Proto-Italic *flagmā, from Proto-Indo-European *bʰl̥g- (“to shimmer, gleam, shine”)."(Wiktionary flamma)
amo is from Latin amo, which Wiktionary gives the etymology
"From Proto-Italic *amāō, of disputed etymology. Possibly from Proto-Indo-European *am-a-, *am- (“mother, aunt”), a lost nursery-word of the papa-type. Compare amita (“aunt”), Old High German amma (“nurse”).
Alternatively, Olav Hackstein and Michiel De Vaan suggest a derivation from Proto-Indo-European *h₃emh₃- (“to seize, to take hold”) via Proto-Italic *amāō (“to take hold”), applying a semantic shift “to take by the hand” > “to regard as a friend” > “to love, to be fond of”." (Wiktionary amo)
That is, there's no evidence that those two words are related.
I'd question the Turkish connection, as well. While English "love" translates into many Turkish words, including âşık(Wiktionary âşık), aşk(Wiktionary aşk) seems to be the core word. aşk is "From Ottoman Turkish عشق ('aşk), from Arabic عِشْق (ʿišq).", whereas ışık is "From Ottoman Turkish ایشق (ışık), akin to Old Turkic [script needed] (yaşuk, “light, sun”), derived from Old Turkic [script needed] (yaşu-, “to shine”). See ışımak. Cognate with Azerbaijani işıq (“light”), Turkmen yşyk (“light”)."(Wiktionary ışık) That is, aşk is ultimately Semitic and ışık is ultimately Turkish. It's easy to find connections if you search for words and ignore the etymology. If sun doesn't work, look for light or flame; sooner or later you'll find a connection.
