The puzzle is easily solved: there is no “i” sound in mangia, it is just a graphic device for denoting the palatal pronunciation of “g”, like in gelato. So, apart from the different vowel following it, the sound is the same /ʤ/.
It's the same for “c”: the palatal pronunciation is denoted by inserting an “i” before “a”, ”o” or “u”: goccia (drop), gancio (hook), ciuco (donkey).
To the contrary, the “hard” sound is denoted by inserting an “h” before “e” and “i”: che (conjunction or pronoun), chi (who), alghe (seeweed), ghiro (dormouse).
So the verb mangiare has no “i” in its root; indeed the future tense is mangerò, mangerai, because there's no need to denote the palatal pronunciation.
Sometimes, however, for etymological reasons (but the real reason is some imposition by grammarians), the “i” is preserved; it's the case of plurals, so camicia (shirt) becomes camicie and ciliegia (cherry) becomes ciliegie. In both cases the “i” still has no sound.
The case of cielo (sky) and cieco (blind) is a bit different: ancient Italian (Tuscan) pronounced this “i”; it's still heard in southern Italian speakers.
In some words, though, the “i” sound is present and it is tonic, like farmacia (apothecary) or genealogia. Of course, in this case, it is pronounced with palatal sound.
A curious case are the adverbs già and giù where there is no diphthong (/ʤa/ and /ʤu/), but they're written with an accent in analogy with più.