Well "manger" is "to eat" in French and "mangiare" in Italian which apparently both come from latin "manducare" so maybe the Romanian word comes from here?

Meanwhile French "manquer" and Italian "mancare" seem to come "From Latin mancus, from Proto-Indo-European *man-ko- (“maimed in the hand”)."

Actually we still have "manchot" in modern French to mean one-armed which I always assumed came from "manche" (sleeve) but seems to actually have a different etymology.

That's all sourced from the wiktionary by the way.

The "man-ko-" term seems to confirm why, as I mentioned at the top, mancar in Portuguese came to mean to limp. To say someone is "manco" is effectively to say they're maimed in some way.

"Sleeve" in Portuguese is "manga", which in turn is also the word for mango, the fruit. The Latin "manica" meant "sleeve", while the name of the fruit originally comes from an Indian language, into English, via the Portuguese.

Etymology is quite the rabbit hole!

In Italian "monco" means missing an arm, which comes very close to it