The thermometer is in something (air, water, let's say water to make the following easier to write) that has a temperature.
It has a particular temperature; the temperature of that water is not the same as the temperature of anything else.
That particular temperature is a reality and we can talk of accurate portrayal of it as "real".
The thermometer is not reporting it accurately, so "the thermometer does not give the real temperature" is correct.
We can also use temperature as an abstract noun, to talk about the concept of temperature more widely. Here though, there isn't a specific "real temperature" to talk of.
"Real" with abstract nouns contrasts a view of it we hold to be genuine with one we don't ("Real love"). Saying it doesn't measure "real temperature" would make sense if we had some theory as to why it's working on a completely different concept of "temperature" to that we hold to be "real".
To express the same thing with an abstract use of temperature we'd need to avoid real and use something like "The thermometer doesn't measure temperature accurately".