Human reproductive, mortality and migration behaviours can often be standardized across countries. However, the universality of population growth laws remains a subject of debate. This study models age-specific population data as a three-dimensional tensor and applies high-dimensional tensor decomposition to uncover macro-level patterns in demographic systems across 200 countries over the past 70 years. The findings reveal that, while disparities in age demographics are widening, most nations follow remarkably similar evolutionary trajectories, differing mainly in the pace of change. A universal transition from the demographic dividend to population ageing is evident, with even labour-abundant regions such as Africa, Asia and South America inevitably facing this demographic shift. By incorporating economic indicators, the study quantitatively demonstrates the coordination between population structure and economic growth, while identifying notable exceptions, such as Gulf states that remain both affluent and youthful, and countries like North Korea, Tunisia, Sri Lanka and Ukraine that are ageing without first achieving significant economic wealth.