In this paper, a terahertz (THz) metamaterial absorber (MTMA), incorporating surface Pythagorean tree fractal resonators, was designed and experimentally fabricated on the flexible substrate of polyethylene terephthalate. The design presented two peaks with strong absorption of more than 97% at 0.49 and 0.69 THz. The dual-band absorption peaks were seen to be shifted with the change in the refractive index of the surrounding medium, with a corresponding sensitivity of 0.0968 and 0.1182 THz/RIU. The spectral shift of the reflection resonance dip was utilized as an assessment index to evaluate the sensing performance of the new structure, and it was found to be 2.08 and 2.98 for the two resonance peaks, respectively. It was observed that the proposed structure acted as an epsilon negative material at the first resonance and as a mu negative material at the second resonance. Further investigations on the electric field, magnetic field, and surface current distributions were carried out to elaborate on the absorption characteristics at various resonance frequencies. The proposed sensor is a highly sensitive MTMA which can be used to investigate the interaction of matter with THz waves.