I have two tools I use to measure my refrigerator's temperature.
- Refrigerator thermometer with multiple remote sensors. Mine has two; it has a base station that says in the fridge, and then two sensors that are placed in different places in the fridge with a wire connecting them. This is useful because refrigerators have issues sometimes where the area under the cold exhaust is TOO cold, or the area further away doesn't get cold enough (either because the airflow is insufficient, or the insulation is not good enough, or the door is left ajar). Having multiple sensors lets you test the different areas, and the model I have keeps track of the recent high/low temperatures.
- Infrared ("laser") thermometer for spot readings. These are really helpful because it lets me look around the fridge for issues immediately. I can see if it's very different between one area and another, or if there is any area that's much too warm.
Most of the time, the airflow is the main issue; whether that airflow is being impacted by "inside the fridge" things (like, too much food), or "inside the pipes" things, like ice formation, either way it's the air moving around that's the problem. Unless your compressor itself is failing, in which case it'll be too warm everywhere.