Concurrent advances across fields such as organoid technology, Microelectrode Arrays (MEAs), neuromorphic computing, and machine learning have given rise to a promising research paradigm: Synthetic Biological Intelligence (SBI). SBI refers to engineered systems in which living Biological Neural Networks (BNNs) are interfaced with hardware and software to perform task-oriented information processing in a closed loop. This emerging technology, while still in its infancy, has the potential to deliver highly efficient performance across both computing capabilities and energy consumption. The early stage of this field underscores the need for reliable multi-scale and cross-domain interaction interfaces to support applications in robotics, biomedicine, signal processing, and neuroscience research. The introduction of standardized, platform- and cloud-integrated BNNs has been an important enabler for the scientific community, improving the accessibility of SBI and facilitating further developments. In this survey, we summarize the innovations that contributed to the emergence of SBI and the first testbed interfaces that enabled its embodiment. While existing definitions establish SBI as the closed-loop integration of living BNNs with hardware and software, these do not provide a unified system-level abstraction for comparing heterogeneous SBI interfaces, mapping design choices, or detailing performance metrics to the different stages of the interaction loop. Thus, this work addresses this gap by introducing an Adaptive Bio-Neural Interaction Architecture (ABNIA), a modular abstraction that integrates encoding and stimulation strategies, neural substrate dynamics, decoding, and feedback control into a common framework for system engineering and benchmarking.