My 2012 Honda civic with 19,000 miles has a blue temp light on dash and it seems to be running fine?
As far as I can tell from the owner's manual, the 2012 Civic doesn't have a traditional temperature gauge. Instead, there is a light (a thermometer and an H) to indicate that the temperature is too high and the car is overheating (meaning you should pull over before you damage the engine), and another light (a thermometer and a C) to indicate that the temperature is apparently too low. It sounds like you are talking about the second one (the manual doesn't say what color it is, but blue would make sense).
Note the word "apparently". It seems unlikely that the coolant would actually be cold after driving for a few minutes, so a more likely scenario is that the temperature sensor isn't working. This wouldn't necessarily cause an immediate problem, but for example, if the car really were to overheat, the sensor might fail to detect it and the high temperature warning light might not come on. In that case, you wouldn't know to pull over and you might damage the engine.
If the same sensors control the fan or other aspects of the cooling system, then this malfunction might make the car more likely to overheat as well.