The MUSE TIMER Survey has obtained high signal and high spatial resolution integral-field spectroscopy data of the inner ~ 6×6 kpc of 21 nearby massive disc galaxies. This allows studies of the stellar kinematics of the central regions of massive disc galaxies that are unprecedented in spatial resolution. We confirm previous predictions from numerical and hydrodynamical simulations of the effects of bars and inner bars on stellar and gaseous kinematics, and also identify box/peanuts via kinematical signatures in mildly and moderately inclined galaxies, including a box/peanut in a face-on inner bar. In 20/21 galaxies we find inner discs and show that their properties are fully consistent with the bar-driven secular evolution picture for their formation. In addition, we show that these inner discs have, in the region where they dominate, larger rotational support than the main galaxy disc, and discuss how their stellar population properties can be used to estimate when in cosmic history the main bar formed. Our results are compared with photometric studies in the context of the nature of galaxy bulges and we show that inner discs are identified in image decompositions as photometric bulges with exponential profiles (i.e., Sérsic indices near unity).