The "Horizon" was a J-Class and the "Fortunate" was a Y-Class; all in the vein of the "Botany Bay" which was a DY-100 class and the "Mariposa" which was a DY-500 class. It would make sense that they likely received 'names' as classes once the Federation was formed, because we are only 10-20 years out from the known period that they were using Daedalus-Class starships. However, it should also be noted that there was reference to a Neptune-class vessel in Season 2, so it would seem 'name'-classes do exist on some level.