i think in italian "coppia" refers mostly to couple of people like husband and wife, because if they said to people not involved with each other they will say pair, so shoes, are definitely not in love with each other so they are not couple. Although you never know

why is "ho una coppia de scarpe" wrong?

I suppose a couple isn't a pair.

A pair of shoes are shoes that match, a couple of shoes needn't be. It's the set way, in English you simply don't say "I need a new couple of shoes for work"