Why is induced EMF of an inductor considered as average voltage and not RMS or maximum?
If you really think about what RMS physically means for inductors and capacitors, it doesn't make a lot of physical sense. RMS was intended for representing the equivalent DC heating in resistors. Average and peak values tends to make a lot more physical sense for inductors or capacitors.
That said, when you do the math you have to keep things consistent; You can't just mix up RMS, average, and peaks.
But your professor must have been speaking about some specific context that you left out or did not notice. For example, if you are concerned about the current in an inductor in a switching converter, where the current ripples above and below a mid-point (rather than weaving above and below 0A), you probably want to work with the average.
If you want to figure out how much charge has been stored in a cap at the end of a cycle, you would work with the instantaneous voltage at the end of the cycle (which would also be the peak voltage if the current was going in one direction into the capacitor the entire cycle so that the voltage only ever climbed) since that voltage represents the history of charge in the capacitor.
Then for that capacitor, if you wanted to find out the equivalent current that would produce that voltage if it were constant, you would use the average current.
You can see from these examples it is all based on context. When in doubt, especially for complex waveforms, you should use instantaneous values. This could be the equation describing the waveform, or actual data points if it is an arbitrary signals measured in the real world. This is what nature does. Nature doesn't know the RMS, the average, or the peak. It doesn't know the equation of the waveform. It just knows the moment-to-moment values. But it is cumbersome do it this way since it involves working with an equation or hundreds if not thousands of data points.
That is the reason we have things like RMS, average, and peak. They are just all abstractions that boil down all those datapoints into a handful of numbers that we humans can use on paper-napkin calculations. But since they are boiled down, they make assumptions and do not capture the entire behaviour of the waveform. They have their limitations and do NOT apply in all cases so you just have to know what you are working with.
Another example of such an abstraction is reactive power and apparent power. In reality, it is all real power. Just that if you have X amount of real power flowing from source to load, and then X amount of real power at some later time flowing back from load to source, it is accounted for as reactive power. But working with it this way requires working with the power waveform as it goes above and below zero (representing changes in direction of power flow) and is difficult to handle without a computer.
If you want a specific answer to what your prof said, please provide us with the context about what was being discussed as the time.

Generally, but not always, upper case notation defines a DC value, DC-bias or a peak amplitude while lowercase represents an AC value (to be superimposed on top of a DC-bias) or instantaneous value (such as that from an equation describing the waveform.