Do I need an overlapping in STFT if I want to play the signal back?
Note that the algorithm in the article uses a windowed FFT for analysis.  Window functions help avoid certain FFT artifacts, but they strongly attenuate any signal at the ends of the window.  So if you don't overlap, almost all the audio (any pitch changes, etc.) at both edges of each window would be lost, and you would end up with a choppy re-synthesized result.  Also, the window function usually peaks in the middle of the window, and is thus produces more accurate analysis there.  So the closer together the window centers, the more accurate the analysis involved in the pitch shift algorithm.  But the window needs to be longer than this distance for accuracy in frequency estimation.  Thus the overlap of 75% instead of just 50% in the article.

Synthesis for playback often involves the phase vocoder algorithm plus cross-fading of overlapped windows to avoid discontinuity ticks and other glitches at the synthesis window boundaries.