Do airline pilots identify failed engines in the same way as light aircraft pilots?
Yes, the dead foot, dead engine does apply (but not as much as you might think), and, yes, airline pilots need to rely on their engine instruments in order to figure out which engine has failed, especially when you're flying a 4-engine aircraft. Actually, and with obvious prejudice, I prefer to say that the Flight Engineer will use the engine instruments to tell the Captain which engine has failed. Couldn't resist that.

Sudden loss of power on 727-100s and 747-100s/-200s is not nearly as noticeable as, say, on a Cessna 310. If you lose the middle engine (#2) on a 727, all you're going to notice is the loss of 1/3 of your power. Even the loss of engines 1 or 3 will not produce much yaw because they're mounted close to the aircraft centerline.

On the transport-category aircraft I flew it was unlikely that your foot would notice an in-flight engine failure because you took your feet off the rudder pedals as soon as you lifted off and didn't put them back on until short final. Adverse yaw was taken care of by the yaw damper.

If the loss of an engine came during descent at flight idle, the pilots might not notice it unless there is an obvious alert of some kind. We had one quit on us descending into Miami once and neither I nor the f.o. noticed it. Our alert was the f.e. saying something like, "We just lost #3." My increased work load consisted of simply saying something like, "See if you can get it running again. If it doesn't light off, forget it."

If you deliberately shut down an engine, you of course know which engine it is. That happened to me once. Again another vote for f.e.s. This one informed me that the oil pressure was dropping and oil temp climbing on one of the engines. He recommended shutting it down, which we did.

Generally speaking, when you lose an engine suddenly without warning when climbing out, I think the thing to remember is to take your time, fly the airplane with the configuration you have, and work the checklist for the situation at hand. That checklist will have procedures to ensure you shut down the engine that has failed, not one that's still operating. In other words, just do what you do in the sim every 6 months.