Poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (P(3HB-3HV)) copolymer’s properties depend on (i) the molar fraction of comonomers, (ii) the overall molar mass, and (ii) the chemical compositional distribution. This work aims at providing a better understanding of the effect of the P(3HB-3HV) molecular structure, produced from mixed cultures and waste feedstock, on copolymer crystallization and tensile properties. Conventional biopolymer characterization methods (differential scanning calorimetry, X-ray diffraction, and polarized optical microscopy) were coupled to both classical one-dimensional (1H and 13C) and advanced two-dimensional (diffusion-ordered spectroscopy (DOSY) and 1H/13C heteronuclear single quantum coherence (HSQC)) nuclear magnetic resonance (NMR) spectroscopy techniques. The obtained results evidenced that (i) a high-quality copolymer could be achieved, even from a waste feedstock; (ii) increasing the 3HV content displayed a positive impact on P(3HB-3HV) mechanical properties only if good interactions between 3HB and 3HV moieties were established; and (iii) the purification process eliminated short-length 3HV-rich chains and promoted homogeneous co-crystallization. Such optimized microstructures enabled the maximal stress and strain at break to be increased by +41.2 and +100%, respectively.