It's nothing more than casual racism.
Laurie's father eloped with an Italian pianist.
Wikipedia covers it quite thoroughly…
Theodore "Laurie" Laurence – A rich young man who lives opposite the Marches, older than Jo but younger than Meg. Laurie is the "boy next door" to the March family and has an overprotective paternal grandfather, Mr. Laurence. After eloping with an Italian pianist, Laurie's father was disowned by his parents. Both Laurie's mother and father died young, so as a boy Laurie was taken in by his grandfather. Preparing to enter Harvard, Laurie is being tutored by John Brooke. He is described as attractive and charming, with black eyes, brown skin, and curly black hair. He later falls in love with Amy and they marry; they have one child, a little girl named after Beth: Elizabeth "Bess" Laurence. Sometimes Jo calls Laurie "Teddy".