Low-power ambient backscatter communication (AmBC) relying on radio-frequency (RF) energy harvesting is an energy-efficient solution for batteryless Internet of things (IoT). However, ambient backscatter signals are severely faded by dyadic backscatter channel (DBC), limiting connectivity in conventional orthogonal time-division-based AmBC (TD-AmBC). In order to support massive connectivity in AmBC, we propose sparse-coded AmBC (SC-AmBC) based on non-orthogonal signaling. Sparse code utilizes inherent sparsity of AmBC where power supplies of RF tags rely on ambient RF energy harvesting. Consequently, sparse-coded backscatter modulation algorithm (SC-BMA) can enable non-orthogonal multiple access (NOMA) as well as M-ary modulation for concurrent backscatter transmissions, providing additional diversity gain. These sparse codewords from multiple tags can be efficiently detected at access point (AP) using iterative message passing algorithm (MPA). To overcome DBC along with intersymbol interference (ISI), we propose dyadic channel estimation algorithm (D-CEA) and dyadic MPA (D-MPA) exploiting weighted-sum of the ISI for information exchange in factor graph. Simulation results validate the potential of the SC-AmBC in terms of connectivity, detection performance and sum throughput.