Water added at room temperature needs time to cool. The dispenser temperature immediately after refilling does not establish the temperature of food already stored elsewhere in the fridge.
What the observation tells you
The tank volume, refill temperature and normal door use affect the observation. Repeatedly drawing off the water and adding more room-temperature water prevents a fair comparison. Lowering the fridge setting sharply can affect food without resolving that loading pattern.
A useful next step
Follow the manufacturer's normal filling guidance and allow suitable cooling. If the water remains unexpectedly warm alongside a measured cabinet problem, describe both conditions when enquiring rather than treating the dispenser alone as a complete cooling test.
Manuals and supporting guidance
Model-specific instructions take priority. These references support the practical principles discussed here; they do not diagnose your appliance.