I checked AirPort Utility, but there is no (Identify feature) in any menu item.
Unfortunately, the symptoms that you are describing likely indicate that you may have a defective AirPort Express.
As a last resort, try a Factory Default Reset as follows:
Power off the Express for a few minutes
Hold in the reset button first and continue to hold it in for another 9-10 seconds while you simultaneously plug the Express back in to power
Release the reset button after the hold period
If the Express is operating normally, you will see a solid amber light for about 30 seconds, then the light will begin to slowly blink amber, indicating that the Express has been reset and is operating normally.
If you see an alternating green/amber light at this point, and you have tried the Factory Default Reset a few times, the Express is defective and will need to be replaced.