AI systems on edge devices require online continual learning -- adapting to non-stationary streams and unfamiliar classes without catastrophic forgetting -- under strict power constraints. We present CLP-SNN, a spiking neural network with a self-normalizing local learning rule and a spike-driven neural state machine for autonomous on-chip learning, implemented on Intel's Loihi 2 neuromorphic processor. On OpenLORIS few-shot experiments, CLP-SNN matches replay-based accuracy rehearsal-free. On Loihi 2, CLP-SNN achieves 113x lower latency (0.33 ms vs. 37.3 ms) and 6,600x lower energy (0.05 mJ vs. 333 mJ) than the strongest edge-GPU baseline. This gain decomposes into algorithmic efficiency (~14.5x latency, ~22.6x energy on the same GPU) and neuromorphic hardware co-design (~7.8x latency, ~295x energy) exploiting event-driven learning and sparse graded-spike communication. We show that co-designed brain-inspired algorithms and neuromorphic hardware can break traditional accuracy-efficiency trade-offs in edge AI.