Sorry Bazile, I have only just seen you post as i seemed tired yesterday so had an early night.
Now,I basically agree with the posts of Mamy and 237....but let us be more specific in your case.
Oddly enough, anal polyps are relatively uncommon, compared with those higher up in the rectum and colon and haven't been studied so thoroughly and are probably less likely to become cancerous..probably.
Now 2006...small fibrous anal polyp....2018, 12 years later the polyp is 2cms......so not much going on there.
Yes, it probably does need removing as if it isn't then you may get rectal bleeding with anaemia, it may become inflamed, it may itch or rarely it may prolapse (come out and won't go back in).Turning cancerous is a possibility but unlikely.
Your problem is the anaesthesia,as it is with many people. it can easily be done under sedation and i would contact the surgeon's secretary, tell her your fears and ask if he/she (the surgeon)will do this polyp removal under sedation though an IV drip.
Yes you could "chicken out" and monitor the situation, but in my opinion with that choice would go a small but present risk of complications as outlined above.