If one needed to replace a wire to a coolant temp gauge, would I be right in thinking it would need to be exactly the same gauge wire to ensure a correct reading?
Not really. Unless you went for ridiculously thin wire, the resistance of it would be negligible, especially as the gauge is drawing very little current.
I believe - maybe @MorkC68 will know - that unless there are environmental factors like heat resistance, you can use normal cable even for a thermocouple.
Normal cable doesn’t have the correct resistance, for example a K type or an N type need the correct compensating cable from thermocouple to your readout. If you mix an N type thermocouple with K type compensating cable you can potentially have a 100 degrees Celsius error (ask me how I know). Your readout may be K type, may be N type, either way it will say. This link from Omega tells you all abou it. https://www.omega.com/en-us/resources/thermocouple-junction-principles