That’s right: one function of the present perfect tense is to “express a state or condition that began in the past and continued up to and including the present.” In the sentence “Monique has stayed here for months,” in particular, the present perfect “has stayed” is used because Monique stayed in that place from sometime in the past up to the present time; at the very moment of speaking, however, Monique no longer stays in that place. If we used the present perfect progressive, “Monique has been staying here for months,” the sense would be that Monique has stayed in that place from sometime in the past up the present, has not left the place, and is expected to stay there in the foreseeable future.
In short, the present perfect is an ended or perfected action at the present time or at the moment of speaking; in contrast, the present perfect progressive is a continuing action that started sometime in the past and is expected to continue beyond the present into the foreseeable future.