Identification capacity has been established as a relevant performance metric for various goal-/task-oriented applications, where the receiver may be interested in only a particular message that represents an event or a task. For example, in olfactory molecular communications (MCs), odors or pheromones, which are often a mixture of various molecule types, may signal nearby danger, food, or a mate. In this paper, we examine the identification capacity with deterministic encoder for the discrete affine Poisson channel which can be used to model MC systems with molecule counting receivers. We establish lower and upper bounds on the identification capacity in terms of features of the affinity matrix between the released molecules and receptors at the receiver. As a key finding, we show that even when the number of receptor types scales sub-linearly in the number of molecule types $N,$ the number of reliably identifiable messages can grow super-exponentially with the rank of the affinity matrix, $T,$ i.e., $\sim 2^{(T \log T)R},$ where $R$ denotes the coding rate. We further derive lower and upper bounds on $R,$ and show that the proposed capacity theorem includes several known results in the literature as its special cases.