Fractional cosmology modifies the standard derivative to Caputo’s fractional derivative of order μ, generating changes in General Relativity. Friedmann equations are modified, and the evolution of the species densities depends on μ and the age of the Universe t_U. We estimate stringent constraints on μ using cosmic chronometers, Type Ia supernovae, and joint analysis. We obtain |$\mu =2.839^{+0.117}_{-0.193}$| within the 1σ confidence level providing a non-standard cosmic acceleration at late times; consequently, the Universe would be older than the standard estimations. Additionally, we present a stability analysis for different μ values. This analysis identifies a late-time attractor corresponding to a power-law decelerated solution for μ < 2. Moreover, a non-relativistic critical point exists for μ > 1 and a sink for μ > 2. This solution is a decelerated power-law if 1 < μ < 2 and an accelerated power-law solution if μ > 2, consistent with the mean values obtained from the observational analysis. Therefore, for both flat FLRW and Bianchi I metrics, the modified Friedmann equations provide a late cosmic acceleration under this paradigm without introducing a dark energy component. This approach could be a new path to tackling unsolved cosmological problems.