The sixth-generation (6G) mobile networks are expected to operate at a higher frequency to achieve a wider bandwidth and to enhance the frequency reuse efficiency for improved spectrum utilization. In this regard, three-dimensional (3D) spatial reuse of millimeter-wave (mmWave) spectra by in-building small cells is considered an effective technique. In contrast to previous works exploiting microwave spectra, in this paper, we present a technique for the 3D spatial reuse of 28 and 60 GHz mmWave spectra by in-building small cells, each enabled with dual transceivers operating at 28 and 60 GHz bands, to enhance frequency reuse efficiency and achieve the expected spectral efficiency (SE) and energy efficiency (EE) requirements for 6G mobile networks. In doing so, we first present an analytical model for the 28 GHz mmWave spectrum to characterize co-channel interference (CCI) and deduce a minimum distance between co-channel small cells at both intra- and inter-floor levels in a multistory building. Using minimum distances at both intra- and inter-floor levels, we find the optimal 3D cluster size for small cells and define the corresponding 3D spatial reuse factor, such that the entire 28 and 60 GHz spectra can be reused by each 3D cluster in each building. Considering a system architecture where outdoor macrocells and picocells operate in the 2 GHz microwave spectrum, we derive system-level average capacity, SE, and EE values, as well as develop an algorithm for the proposed technique. With extensive numerical and simulation results, we show the impacts of 3D spatial reuse of multi-mmWave spectra by small cells in each building and the number of buildings per macrocell on the average SE and EE performances. Finally, it is shown that the proposed technique can satisfy the expected average SE and EE requirements for 6G mobile networks.