Age estimation from finger vein images has been widely considered impractical due to severe demographic biases in public datasets and physiological confounding factors like gender. To overcome these limitations, we propose MAGE-Vein, a novel multi-instance, multi-task learning framework. Our approach extracts robust structural aging signs by employing a hybrid feature-level fusion of three fingers, effectively suppressing local imaging noise. Furthermore, simultaneous optimization of gender classification conditions the network to effectively eliminate gender-specific vascular variations. Evaluated on a demographically balanced dataset of 402 subjects, MAGE-Vein achieves a mean absolute error of 6.12 years and a correlation of 0.880. Our results not only overturn the conventional consensus regarding the limitations of the finger vein modality but also demonstrate that previous estimation failures were primarily artifacts of biased public datasets.