It's puzzling to be asked the question you've asked: Was the hug you felt with your deceased mother real or not? I could answer "yes" or "no" and justify both. Your brain does know enough of how hugging her felt to reproduce it in a dream. But that facility with "presentification" (kinaesthetic recreation) along with your other unusually vivid dream skills suggests that you have unusually strong and frequent contact with your mother "on the other side" (I only use quotes because I don't know what else to call it.) A woman I knew in her late twenties who had that ability with her deceased brother found that with training she could develop it into psychic reading and channeling skills. This ability to "pierce the veil" between the here and the hereafter is a mixed blessing fraught with significant responsibilities if one pursues it.
It's also important to consider how long you want to keep up regular contact with your deceased mother, because that has an effect on where you are living most intensely with your emotions and goals. You're still in the first year of bereavement, so it's not unusual that your connection with her are strong. In my experience it is normal for the Higher Power that Directs our dreams to design them symbolically to help the dreamer pass through the significant and introverted bereavement experience so that she can come out the other side ready to continue her life without her mother (most of the time) but with significant gains made thru incorporating the relationship now completed. That's not to suggest that everybody should leave behind such a connection to the hereafter to live solely in this world, until they're nearing their own end.
So to your question "Was her hug real?" I respond, "Can you be satisfied that it WAS real and still let her go? Or is keeping her real important enough to you to want to lead part of your life in communication with the "other side," even if that might mean experiencing less of the world on this side during your lifetime?"
There's a lot of wonderful uses you can put your dreams to, such as divine guidance, artistic inspiration (since dreams are artworks just emerging from the unconscious of our species), psychological and/or psychic expertise (ie becoming a counselor or college-trained psychotherapist), understanding aspects of inner and outer reality few others in materialistic societies ever know. etc. For many of these uses you need to study and get serious about keeping track of your dreams. but you'll never be without the awe and wonder that you have now--I've been at it for 54 years and I didn't even realize I was getting spiritual guidance through my dreams, whether I remembered all of them or not, until 12 years or so into my dream-fueled path, when I began my second masters degree in Clinical Psychology @ 30.