This paper is devoted to the numerical analysis of a fully discrete finite element approximation for the stochastic Benjamin-Bona-Mahony equation driven by multiplicative noise. We first establish the existence and uniqueness of solutions to the stochastic BBM equation within an appropriate variational framework and derive several stability estimates for the continuous problem, including an exponential stability result. For the numerical approximation, a conforming finite element method is employed for spatial discretization and is coupled with the implicit Euler-Maruyama scheme for time integration. The convergence of the fully discrete scheme is investigated under two different classes of multiplicative noise. When the noise coefficient is bounded, we obtain optimal strong error estimates in full expectation by combining exponential stability properties of both the stochastic BBM solution and its fully discrete counterpart with a stochastic Gronwall inequality. In the case of general multiplicative noise, where boundedness assumptions are no longer valid, a localization technique based on high-probability events in the sample space is introduced, leading to sub-optimal convergence rates in probability. Finally, numerical experiments are presented to corroborate the theoretical results and to demonstrate the performance of the proposed method.