Nah, the war is going to be over in the 1860s so the labels will be there by the 1870s at the latest. It isn't going to take two generations to start labeling factions.
Labeling them, probably not. Factions actually having any formal organisation and accepting those labels... it's a larger step than you might think, and one of the main drivers for it is universal suffrage - something the CSA will not even have for white males IIRC. The Democrats had no real external competition and often fierce internal contests in the OTL South, yet I don't think there were real organised factions within them. The lack of a viable national opponent will accelerate the process, but it's not going to happen overnight. France didn't have real parties before about 1900 (before then it's mostly guesswork and back-projection) and the right only really started having them in the Fourth Republic.
The Presidential system may accelerate this, due to the single national candidacy, but governorships don't seem to have OTL on a state level, and it seems plausible you could get something probably more like the US system was intended to be, where the House decides between multiple candidates every six years.