Algorithms designed for addressing typical supervised classification problems can only learn from a fixed set of samples and labels, making them unsuitable for the real world, where data arrives as a stream of samples often associated with multiple labels over time. This motivates the study of task-agnostic continual multi-label learning problems. While algorithms using deep learning approaches for continual multi-label learning have been proposed in the recent literature, they tend to be computationally heavy. Although spiking neural networks (SNNs) offer a computationally efficient alternative to artificial neural networks, existing literature has not used SNNs for continual multi-label learning. Also, accurately determining multiple labels with SNNs is still an open research problem. This work proposes a dual output spiking architecture (DOSA) to bridge these research gaps. A novel imbalance-aware loss function is also proposed, improving the multi-label classification performance of the model by making it more robust to data imbalance. A modified F1 score is presented to evaluate the effectiveness of the proposed loss function in handling imbalance. Experiments on several benchmark multi-label datasets show that DOSA trained with the proposed loss function shows improved robustness to data imbalance and obtains better continual multi-label learning performance than CIFDM, a previous state-of-the-art algorithm.