How does the pilot check if an engine fire is real? I've read this question and its answers basically asking why putting out an engine fire is not left to the computer but a manual task. 

One of the points mentioned is that there simply might be a sensor error or similar and the engine isn't actually on fire, in which case you obviously don't want the computer to needlessly shut down an engine. 

My question however is, how do the pilots assess wether the engine is actually on fire or not? 

All planes I've been on had their engines very close to the aircraft and I doubt they are easily if at all visible from the flight deck. I would assume a good indicator might be the loss of thrust, but that could also be detectable by a computer. Do the pilots have cameras they can activate to check on the engines, or do they call the cabin personal and ask them to take a peek, do they walk out and check or is it based on how loud the passengers scream in the back? I guess all of these work, but what is the common procedure?
Engine fire implies that there is a strong heat source (commonly a fire, but could be a hot substance) in a location there should not be, usually around the outside of the engine.


Engine fires do not necessarily need to be connected with a loss of thrust.
Visually, they are pretty diverse- there may be smoke, there may be little smoke (aside from that, what would you do at night?).


The general rule is to treat the cockpit instrument indications as real every time. You do not need a visual confirmation.

You would rather take safe action a thousands times too many than one time too little. The aircraft is designed to fly on one engine and it's probably one of the most common exercises for pilots in simulators.

You are correct that sensors do get it wrong. Here is a very recent incident that involved just that. The reason for the sensor discussion is that a decision is only as good as the information you can base it on, and a human is generally a better decision maker than a computer as it currently stands. Take this very complicated and confusing situation for instance- how would a computer decide and respond?

There is no physical damage in hitting the fire button which will shut the engine down. Discharging the Halon fire agents would normally require a new bottle. It is a bit of a 'hard' engine shutdown: it cuts fuel, bleed air, electricity and hydraulics. If the fire warning (i.e. heat source) persists, you generally escalate it to discharging fire agents, although in most cases shutting the engine off is a sufficient measure.