Homeostasis relies on a precise balance of fate choices between renewal and differentiation. Although progress has been done to characterize the dynamics of single-cell fate choices, their underlying mechanistic basis often remains unclear. Concentrating on skin epidermis as a paradigm for multilayered tissues with complex fate choices, we develop a 3D vertex-based model with proliferation in the basal layer, showing that mechanical competition for space naturally gives rise to homeostasis and neutral drift dynamics that are seen experimentally. We then explore the effect of introducing mechanical heterogeneities between cellular subpopulations. We uncover that relatively small tension heterogeneities, reflected by distinct morphological changes in single-cell shapes, can be sufficient to heavily tilt cellular dynamics towards exponential growth. We thus derive a master relationship between cell shape and long-term clonal dynamics, which we validated during basal cell carcinoma initiation in mouse epidermis. Altogether, we propose a theoretical framework to link mechanical forces, quantitative cellular morphologies and cellular fate outcomes in complex tissues.