This article investigates semantic communication in high-speed mobile Internet of Vehicles (IoV), focusing on spectrum sharing between vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communications. We propose a semantic-aware spectrum-sharing (SSS) algorithm using deep reinforcement learning (DRL) with a soft actor-critic (SAC) approach. We start with semantic information extraction, redefining metrics for V2V and V2I spectrum sharing in IoV environments, introducing high-speed semantic spectrum efficiency (HSSE) and semantic transmission rate (HSR). We then apply the SAC algorithm to optimize decisions V2V and V2I spectrum-sharing decisions on semantic information. This optimization aims to maximize HSSE and enhance the success rate of effective semantic information transmission (SRS), including determining the optimal V2V and V2I sharing strategies, transmission power, and the length of transmitted semantic symbols. Experimental results show that the SSS algorithm outperforms other baseline algorithms, including other traditional-communication-based spectrum-sharing algorithms and spectrum-sharing algorithm using other reinforcement learning approaches. The SSS algorithm exhibits a 15% increase in HSSE and approximately a 7% increase in SRS.