In streaming recommender systems, the traditional approach for handling new user IDs or item IDs is to assign randomly initialized ID embedding, leading to two practical issues: (i) Items or users with insufficient interactive data can result in suboptimal prediction performance; and (ii) The embedding of new IDs or low-frequency IDs will consistently increase the size of the embedding table, thereby consuming unnecessary memory. To this end, we propose a reinforcement learning-based Automatic Shared Embedding Assignment framework, AutoAssign. To be specific, an Identity Agent serves to (i) field-wisely represent low-frequency IDs by utilizing a small number of shared embeddings, so as to enhance the embedding initialization; and (ii) dynamically identify the ID features that need to be retained or eliminated in the embedding table. We conduct extensive experiments on three public benchmark datasets and observe that AutoAssign can significantly improve the recommendation performance by alleviating the cold-start problem. Besides, AutoAssign reduces the memory space by 20-30 %, which demonstrates the effectiveness and efficiency of our framework in practical streaming recommender systems.