Any unique characteristics to identify aircraft engine?
Some specific aircraft-engine combinations can be identified, but the engines themselves, no, because you don't see most of them—the nacelle is designed for each aircraft-engine combination. For example


  
  The CFM56 on a Boeing 737NG is easy. It has a flat bottom.
  


the CFM56 on a Boeing 737NG is easy mainly because it is the only engine type that aircraft uses. And the engines from the same series when used on A320-family don't have flat bottom!

For A320-family, the CFM56 engines use separate exhausts, with shorter nacelle and hot section jet extending behind it, while the IAE V2500 ones use mixed exhaust with long nacelle, so you can tell them apart on those aircraft. But on NEO both engine types used have separate streams, and the only visible difference is the slight one in shape of the blades.

It is actually quite common for aircraft to only use one engine type, and using more than two types is rare, so most of the time if you can tell the aircraft, including variant, you should already know.