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Lane changing and lane merging remains a challenging task for autonomous driving, due to the strong interaction between the controlled vehicle and the uncertain behavior of the surrounding traffic participants. The interaction induces a dependence of the vehicles' states on the (stochastic) dynamics of the surrounding vehicles, increasing the difficulty of predicting future trajectories. Furthermore, the small relative distances cause traditional robust approaches to become overly conservative, necessitating control methods that are explicitly aware of inter-vehicle interaction. Towards these goals, we propose an interaction-aware stochastic model predictive control (MPC) strategy integrated with an online learning framework, which models a given driver's cooperation level as an unknown parameter in a state-dependent probability distribution. The online learning framework adaptively estimates the surrounding vehicle's cooperation level with the vehicle's past trajectory and combines this with a kinematic vehicle model to predict the probability of a multimodal future state trajectory. The learning is conducted with logistic regression which enables fast online computation. The multi-future prediction is used in the MPC algorithm to compute the optimal control input while satisfying safety constraints. We demonstrate our algorithm in an interactive lane changing scenario with drivers in different randomly selected cooperation levels.