I’m amazed this hasnt been answered correctly yet. No! They are not completely compatible. No! You should not replace a 5C battery with a 5S battery.
The connector is the same and the 5S has slightly higher capacity, which means slightly longer battery life. HOWEVER the 5S battery is slightly wider which means it either will not fit or will be a dangerously tight fit. They also expand with wear which could lead to serious problems even if it initially fits.
Remember: If these batteries puncture they will catch fire and potentially explode.
I’ve measured 5S batteries to be between 33 and 34mm wide, whereas OEM 5C battery cells should only be 32mm wide.
I highly discourage using 5S batteries in 5C’s even if it initially fits! You do not want to burn your house down (or somebody else’s)!