We study the coupled dynamics of multilayer networks with symmetric (MLs) and asymmetric (MLas) inter-layer connections. The symmetric inter-layer connections arise from a one-to-one correspondence between the nodes of different layers. In contrast, asymmetry results from the multiplicity of inter-layer connections, achieved by randomizing the links while preserving their overall density, thereby allowing one-to-many inter-layer connections. We investigate how different types of inter-layer coupling impact the dynamics of non-identical multilayer networks. We find that the specificity of one-to-one inter-layer connections facilitates intra-layer synchronization (ILS). In contrast, for networks with random inter-layer connectivity, ILS depends on how randomness affects intra-layer homomorphism (the set of permutations that preserve the network structure). Furthermore, amplitude death (AD) in MLs is observed at lower connectivity strength and frequency mismatch than the MLas. Moreover, AD in MLs depends on the density and topology, but does not depend on the size of the networks. On the other hand, AD in MLas is influenced by network size in addition to density, topology, and inter-layer mismatches. Moreover, both the MLs and MLas exhibit multi-stability, with the faster layer exhibiting a remanent periodic phase-locked oscillation, irrespective of the topology and inter-layer connectivity. In addition, remnant synchrony between nodes with homomorphic relationships is observed in the slower layer. Overall, we propose that symmetric inter-layer connections should be preferable for achieving intra-layer synchronization-regardless of global synchronization-and for sustaining permanent memory in multilayer networks with mismatched nodes across layers. However, to mitigate AD at low coupling values and layer mismatch, asymmetric inter-layer connectivity is more advantageous.