Can we supply 2A DC current at 5V in the pair of ethernet CAT 5e cable?
CAT5e cable has 22AWG (at most) wire in it and could have 26 awg, this means that the cable has a resistance of up to 16 ohms per 1000ft. A 10ft section would have 0.16Ω, at 2A this would cause a loss of 0.32V one way, with a total voltage drop of 0.6V.
This would probably not be enough voltage to run your device as it most likely requires 5V, not 4.4V. The other problem would be that the voltage drop would fluctuate with the current causing problems for the device.
With 26 AWG the problem gets worse as it's resistance is 40Ohms per 1000ft and would cause an even bigger drop.
So the answer is, probably not it depends on the length and the wire gauge of the cable. If you do it would be best to have a regulator on the device end or some other way to keep the voltage above 5V.
POE works by scaling up the voltage and running low current and by running AC power, not DC.