Of course much depends on the definition of what a sense is. The five senses that are usually referred to, sight, hearing, touch, smell and taste have specific sensing organs - receptors that are to a certain degree specific to some stimulation, for example chemical, or pressure. In fact there are more senses, such as nociception, or thermoception. In addition, there are senses that deal with stimulation inside of the body. Emotions in contrast seem to be aroused by the results of other senses, something we see, or hear - or imagine seeing or hearing. Thus, there seems to be a big difference. Emotional processes (let's not go back to the definition issue for the time being) follow other sense data, are not dependent to a specific modality, and, we now know, are linked to relatively complex networks in the brain that are not modality specific.
But is intuition not "the sixth sense"? Well technically not. More often sixth sense is associated with the notion of paranormal phenomena, such as telepathy. Independent of the fact that there is no scientifically validated evidence of such paranormal phenomena, the use of the term sense in this context is colloquial.
But what could it then be? I am at this point wondering whether augmented reality is a useful metaphor for the relationship of emotions and perception. Emotion can provide a different dimension to sense data - the dark ally that feels creepy, the cake that looks yummy, the dog that looks dangerous. In all of these examples words are used to convey a feeling that is the summary function of a rapid and typically automatic evaluation of properties of the object(s) perceived in relationship to their implication. One aspect of emotions is that they send nonverbal messages that are shorthand for the implication. They may trigger in human verbal associations and interpretations, but they need not. Important: these messages need not be consciously received. After all, part of the process is the rearrangement of resources to deal with situations, and that is the case whether one is aware of it or not.
Wait a moment ...? Unconscious emotions? Yes, If you did not already do so, please check out
Winkielman, P., Berridge, K. C., & Wilbarger, J. L. (2005). Unconscious affective reactions to masked happy versus angry faces influence consumption behavior and judgments of value. Personality and Social Psychology Bulletin, 1, 121-135.