How does the pilot check if an engine fire is real?
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.