Fully homomorphic encryption (FHE) enables computations on encrypted data. Existing FHE schemes are primarily categorized into RLWE-based word-wise schemes and LWE-based bit-wise schemes. Efficient combination of different FHE schemes adapt to real-world application has emerged as a research focus. This paper proposes UniFHE, the first FHE accelerator that support diverse algebraic structures using general arithmetic units to enable higher performance. UniFHE is compatible with both RLWE-based and LWE-based FHE schemes without modifications to original ingenious algorithm design. To support both finite ring and complex field operations, UniFHE introduces the general arithmetic unit and further constructs core computation structures. To balance on-chip memory demands of different schemes, UniFHE adopts a multi-pipeline architecture for LWE-based schemes. The core functional units for RLWE-based schemes are spliced based on the LWE-based pipelines. Additionally, on-chip plaintext encoding mechanism significantly reduces off-chip memory bandwidth demands. Experimental results show that, beyond superior area and energy efficiency, UniFHE delivers up to 13.6× higher performance compared to scheme-specific accelerator combinations. Additionally, in hybrid schemes, UniFHE achieves a speedup of 3.2$\times$ over the state-of-the-art general FHE accelerator Trinity.