We develop a minimal model to describe growing dense active matter such as biological tissues, bacterial colonies, and biofilms, which are driven by a competition between particle division and steric repulsion. We provide a detailed numerical analysis of collective and single-particle dynamics. We show that the microscopic dynamics can be understood as the superposition of an affine radial component due to the global growth, and of a more complex nonaffine component that displays features typical of driven soft glassy materials, such as aging, compressed exponential decay of time correlation functions, and a crossover from superdiffusive behavior at short scales to subdiffusive behavior at larger scales. This analogy emerges because particle division at the microscale leads to a global expansion, which then plays a role analogous to shear flow in soft driven glasses. We conclude that growing dense active matter and sheared dense suspensions can generically be described by the same underlying physics.