Bosch cordless drill with burned MOSFET
The good (at least not burned part) MOSFET in your pictures I take it is the 04NG which is an N-channel MOSFET.  The added numbers on this parts markings have to do with various information like manufacture, location of manufacture, date or production line numbers.   

As far as it being identical, I would suspect that not to be this case.  If they both were connected to the same location for the heavy sink buss terminal of that part then more likely they would be, but as seen they are connected to two different locations.  If you trace the blue wire and the white wire they likely go to opposite polarities on the battery.  This could be one is used forward and one reverse.   

The one on EBAY could possibly replace the one that does not look burnt.  If the burnt one is the same yes it could work if it is instead the P-Channel version then no it will need the P-Channel version of 04NG.   The issue I see is parts sources like Mouser list 04NG as 40V but most are shown as 120amp units.  The EBAY unit is listed it as 30amp.  I would look for at least a 120A version for replacing.

There is always a chance other parts were damaged, it is a roll of the dice after a tech inspects for obvious parts and replaces those.  Before buying and replacing the obvious components check and see if there is a chance to buy the whole board cheaply.  If the board is close to the part only price you then can reduce the worry on some of the other parts on the board being bad.