Question

What is meant by the statement, "a thermometer measures its own temperature"?

Answer 1

It means it will only measure the temperature of the surrounding (what you expect from a thermometer) under certain conditions : namely that it is at true thermodynamics equilibrium with the surrounding.

There can be cases of steady states, when the Sun shine on a thermometer for instance, where the temperature read by the thermometer does not represent anything but its own temperature. In that case, different thermometers would give different readings.