Fingida translates to "feigned" or "fake." It's a less common, more evocative word than simply saying something is "not real."
In the song, the lyric "Con la sonrisa fingida que me dejó tu partida" (With the feigned smile that your departure left me) powerfully conveys the depth of the singer's pain. It suggests a forced happiness, a mask worn to hide the true sorrow caused by a loved one's absence, making it a poignant and memorable choice.
"Durmiendo Con La Luna" is Elefante's account of a breakup that hasn't ended, it's just gone quiet. The narrator keeps repeating "Aquí estoy" [Here I am] like a check-in nobody asked for, watching the seconds and minutes pass while wearing "Con la sonrisa fingida" [With a fake smile] the split left behind.
The title comes from the closest thing the song has to a picture of that loneliness: "Durmiéndome con la luna" [Falling asleep with the moon], "Despertando con el sol" [Waking up with the sun] — alone through the whole cycle, day into night into day. By the last verse he's "Curándome las heridas" [Healing my wounds], which is as far as the song lets him get: not over it, just further along.