Prior to the temperature light illumination did the Caution System work properly? The Caution System should, key on, not started, illuminate the temperature, overrev, oil and check engine light for 2 seconds accompanied by the buzzer. This is a self check.
When was the last time the water pump was serviced?
If you are experiencing an overheat condition the caution system should have a code flashing and limit the engine revs, yours isn’t. My service manual(2009 DF90A) doesn’t mention anything about the light staying on without an alarm.
Post #2 is a valid point however the ECM controls the warning lights. The ECM is expecting the sensors to fall into a preprogrammed range, if they don’t then the caution system is activated.
A genuine Suzuki Service manual will have the testing procedures for the temperature sensors. Depending on the year Selfixer,com might have a pdf service manual for cheap.