Consumer-grade rechargeable C’s and D’s are just AA’s in a shell, with a higher price tag. Check the mAh ratings, it’s a scam!
In decades past, C’s (and D’s) were more popular, and it made sense to manufacture them separately. So, alkaline and carbon C’s are truly C’s, with higher capacity than their AA counterparts. But their sales volume has fallen off so sharply since the advent of NiMH chemistry, manufacturers just produce tons of the AA cores and wrap 'em differently.
You can find “real” ones, with appropriate capacity (~5AH for C’s, ~10AH for D’s) but they’re super expensive ($5/$10 ea) and usually a few generations behind in terms of chemistry.
Also, good luck finding a charger for the poor things. You could make an upsize adapter for a common AA/AAA charger, but the charge termination logic won’t match the huge capacity – either the -dV-detection will fail because the charger isn’t putting enough current into the cell for the sag to be noticeable, or the overcharge timeout will elapse while the cell is still happily charging along.
My verdict: Just get a set of adapter sleeves and use the NiMH AA’s you already own. Modern NiMH AA performance is on par with archaic carbon C’s and D’s anyway.
(Alkalike C’s and D’s still hold an edge in some ways, but devices that need that much power are going over to Li+ 18650’s anyway, so it’s largely moot.)