Master Romancer is an award obtained by having seven of a species of Piñata resident in your Garden at the same time. This is possible by one of three ways or a combination of the three ways. The most obvious way is by romancing them, however romancing is not required to obtain this award. A second way is buying from the Paper Pets store. The third way is to acquire them from Gretchen Fetchem's.
Not all piñatas have a Master Romance award to obtain. The only exceptions (in terms of species) which this award cannot be obtained for are all of the sours, which cannot be romanced as they are wild, non-resident piñatas, and the Dragonache and Choclodocus, who both cannot be romanced.
Obtaining this award for a piñata will give you the ability to change your label to have the icon for it in the label. In Viva Piñata: Trouble in Paradise, getting the award also allows you to ban the species using the Bouncer Board decoration placed outside the garden.