The coolant temperature gauge in the BMW E30 instrument cluster is connected to the brown-plug coolant temperature sender(12621710511).

The coolant temperature sensor

The temperature sensor is an NTC thermistor, its resistance is higher when the temperature is lower.
It was inserted into a container with liquid silicone that was subjected to freezing and heating. Temperature values were recorded. They are lower that the temperature range of the engine, I may try to measure the higher temperatures later and recalculate the results.

Temperature	Resistance
3°C (38°F)	1300 Ω
25°C (77°F)	620 Ω
60°C (140°F)	186 Ω

The Steinhart–Hart equation coefficients:
A = 0.0021390576483277567
B = 0.00011856786248721395
C = 0.000001702703922601429

Coolant temperature gauge values

The values for the reference marks, they start from 0. The supplied voltage is 5 Volts.

Mark	Resistance	Temperature	Input voltage
0 (blue)	160 Ω	64°C (147°F)	3.1 V
1	85 Ω	82°C (180°F)	2.3 V
2	54 Ω	94°C (201°F)	1.8 V
3 (middle)	35 Ω	106°C (222°F)	1.33 V
4	25 Ω	114°C (237°F)	1.05 V
5 (red)	20 Ω	120°C (248°F)	0.86 V
6	12.5 Ω	132°C (270°F)	0.6 V