This trophy requires that you complete each request on Classic mode.
Classic mode usually became available for me after completing a request on Katamari Forever and Katamari Drive with scores of 100 on both. This was not always the case, though, as sometimes Classic mode became available with, say, 100 on Katamari Forever and 75 on Katamari Drive. And sometimes I'd have 100 on both and have to complete the request again on either Katamari Forever or KatamariDrive, at which point Classic mode would become available.
The following requests do not have Classic mode:
The King- The King's Dream
RoboKing- Tutorial
RoboKing- Katamari Sprinkler
RoboKing- Farewell!