Never seen a nose like mine; is it possible to fix it? (Photos)

Based on your photos, it appears that you have a "hanging columella".  This is usually excessive septal cartilage that pushes the columella, or the cartilage/skin between the nostrils, down. You also may have slight nostril arching, exacerbating the issue on profile.  These issues can be improved with rhinoplasty.  You might consider having someone perform digital "morphing" to show you what might be possible with surgery.

I have seen plenty of noses like yours :)!  What you are calling too much nostril is called a hanging columella or excessive columellar show.  You also have alar notching because of weakness of the cartilages that make up the tip of the nose.  There are several maneuvers that can help this problem.  Trimming of the caudal septum, removing some of the lining inside the nose, and grafting of cartilage to the weaker portion of the tip, and sometimes even composite grafting.   What would help most is a thorough exam and consultation to discuss your options.

Your columella appears to hang on profile view, and this excessive nostril show may be addressed during rhinoplasty surgery, depending on your examination. Hope this helps you. Dr Joseph

Hello and thank you for your question. Based on your
photograph, you may benefit from alar batten grafts to help reduce your alar flare. The most important
aspect is to find a surgeon you are comfortable with. I recommend that you seek
consultation with a qualified board-certified plastic surgeon who can evaluate
you in person.
Best wishes and good luck.
Richard G. Reish, M.D.
Harvard-trained plastic surgeon

Yes, it's possible to decrease your nostril size. From the two pictures you submitted, I think decreasing the "columellar show" ( the skin in between the nostrils is hanging down) would help you the most. Yes you could elongate the nostril edge but that is much more difficult than pulling up the columella. I think you would have a great result and be very happy with that kind of rhinoplasty.