Why does the temp gauge indicate the car is running hot when it is not?

Most likely a bad instrument cluster.
This is a very common problem among GM Products from 2002-2006.
I myself have replaced many of these.
There is people online that fix them.

Btw, I’m available to help over the phone in case u need at https://www.6ya.com/expert/humberto_e9f4330d09229a93

Hi Sandra:
Let's try to figure out the temp guage situation.........
There are 2 parts to this system:
1) - the temperature sensor attached to the engine block (old school) or
- the Engine Coolant Temperature (ECT sensor) Typically attached to the radiator (New school where the car gets to think it knows more than you do), and
2) The Engine Temperature Guage located on the dash where you get to look at it. This could also be a Hot Engine light.
Let's think old school and assume that you have a sender and a guage.....
If the Guage is telling you that the engine is overheating and you KNOW that it is not, then:
- either the sensor is not sending correct information, or
- The guage is not accurate. When the engine is cold and you first turn the ignition key on, what does the temp guage do? As the engine warms up, does the guage suddenly go way up?

Based on too many years experience, I tend to not trust the sender on older vehicles. Try disconnecting the electrical lead from the sensor and see what reaction you get from the guage.
It may be that replacing the sensor will fix the problem.

Ahh ... the gas gauge / fuel level gauge does not work, either.

Maybe you need to replace the motors that are behind those dials. People are getting charged $300-$500 to have the fuel gauge replaced when the problem really is the little motor - 5 or 6 of them for less than $30. 2003 Buick LeSabre owners are having the same problem with their fuel gauges being erratic.