The increasing demand for deep neural inference within database environments has driven the emergence of AI-native DBMSs. However, existing solutions either rely on model-centric designs requiring developers to manually select, configure, and maintain models, resulting in high development overhead, or adopt task-centric AutoML approaches with high computational costs and poor DBMS integration. We present MorphingDB, a task-centric AI-native DBMS that automates model storage, selection, and inference within PostgreSQL. To enable flexible, I/O-efficient storage of deep learning models, we first introduce specialized schemas and multi-dimensional tensor data types to support BLOB-based all-in-one and decoupled model storage. Then we design a transfer learning framework for model selection in two phases, which builds a transferability subspace via offline embedding of historical tasks and employs online projection through feature-aware mapping for real-time tasks. To further optimize inference throughput, we propose pre-embedding with vectoring sharing to eliminate redundant computations and DAG-based batch pipelines with cost-aware scheduling to minimize the inference time. Implemented as a PostgreSQL extension with LibTorch, MorphingDB outperforms AI-native DBMSs (EvaDB, Madlib, GaussML) and AutoML platforms (AutoGluon, AutoKeras, AutoSklearn) across nine public datasets, encompassing series, NLP, and image tasks. Our evaluation demonstrates a robust balance among accuracy, resource consumption, and time cost in model selection and significant gains in throughput and resource efficiency.