As long as the dielectric constant of the different types of oil is similar, you won't find a noticeable difference in capacitance. As
this article shows (referenced before), you will see adifference in capacitance if the dielectric constant of the oil changes. In that case the degradation of the oil with time changes the dielectric constant. This is used as an indicator fopr the quality of the oil.
Also: Whatever method you use, make sure the capacitance of the sensor is a noticeable percentage of the capacitance of the whole setup. If the sensor's capacitance is small, but the setup (wires etc) has a high capacitance, any change in capacitance of the sensor will be seen proportionally smaller on the meter.
2 Examples:
1) Assume a sensor capacity that is 50% of the capacity of the whole test setup. If the capacity of the sensor changes by 10%, the whole setup's capacity changes by 5%. You should be able to measure that.
2) Assume a sensor capacity that is 10% of the capacity of the whole test setup. If the capacity of the sensor changes by 10%, the whole setup's capacity changes by only 1%. This will be difficult to detect.