"Livia has a plate and a cup."
Translation:Livia patellam et poculum habet.
It's a structure called the dative of possession; it's common with the verb "to be" (hence the use of sunt ).
The thing(s) possessed--here, the plate and cup--are the subject of the verb "to be." The person who possesses them (Livia) is put into the dative.
It's by no means a rare or unusual construction.
"I have a brother." Mihi est frater. "My name is Marcus." Mihi nomen est Marcus.
(There's no "have," because instead of saying "I have a brother," we're saying "A brother is to me" / "For me, there is a brother.")