In today's world everyone is under some kind of emotional stress, be it from hard work, bad bosses, family troubles, or health troubles. This kind of dynamic pace of living is no joke, and this is why it is important to have someone you can lean on emotionally at the end of the day. Someone that will at least take your mind away from bad things to allow you to wind down so that you don't have burnout or an anxiety attack.
In order to be emotionally supportive of your partner you actually have to have emotions. Because of this fact if you feel that your significant other is not emotionally supportive, this can mean that your partner is either a psychopath, sociopath, emotionally immature or is afraid of emotions to the point of self-repression. It can just plain mean that your partner never experienced the emotion that is in question so because of that they can't identify it properly.