Can anyone explain why exactly an "Overshooting" phenomena is observed when the fundamental harmonic is removed as seen on the figures? Is it technically right to call this "overshooting" at all ? If not, how is it referred to ?
As already mentioned by @Fat32 in the comments to your question what you're observing is not commonly referred to as overshoot.
As there are no units on both plots comparing the results in both figures is somewhat difficult. The following assumes that the horizontal axes in both plots are just the indices of either the time or frequency domain values (otherwise the graphs cannot be directly related without additional information and/or do not make sense).
The exact effects of what you're observing will generally depend on the frequency, amplitude and relative phase of the harmonics present in your signal.
A rough visual comparison of the green and blue lines in the first plot indicates that indeed the first harmonic has been removed. The difference between the green and blue lines is roughly a sine wave (phase zero) that fits into the window (i.e. no difference at the start, end and 1/2 of the window; largest difference at roughly 1/4 and 3/4 of the window).
Note that if the amplitude and/or phase of the first harmonic are different the time domain signal, resulting after subtraction of the first harmonic, will probably look completely different.
I assume you used a filter to remove the harmonic. What you see in the second plot is the stop band ripple of this filter. If you extended the time signal beyond the orginal length, you would also see a slight smudging, also caused by the filter. This smudging would be extreme, if you just manually set the frequency bin of the harmonic to zero, as this corresponds to a convolution with a very broad si-like function. If you want to go into detail, get a standard text book about signal processing, like "Signals, Systems and Inference" by Alan Oppenheim.