It's not funny for you, but I laugh anyway. I truly think it's a phase. When my son was in the years 2-3 1/2 he CONSTANTLY went by nicknames that he himself made up. Let me think. There was "Greg Wiggle" ,"Anthony Wiggle", "Little Grandpa", "Little Daddy", etc. He would look right at me and say "My name isn't Alex." He would not respond until I finally called by his "name". It annoyed me too, but now I realize that he was just beginning to learn how to use his imagination and playing.
Considering I come from a family were EVERYONE goes by a nickname of their name, I'm amazed at all the angry moms on here. Wow. I go by a nick of my real name, as does my brother, both parents, my own child, who goes by a nickname of his MIDDLE name, my husband, and both his parents. I also work in daycare and the nickname giving is a sign of love and affection, so I don't know what the anger is all about. All of our babies have nicknames, and the children's parents start to use them too. BUT we also use the child's real name most of the time, and save the nickname for a moment of one-on-one playing with them. If our little 10 month old responds to her birth name and nickname, I really think your son knows his name, and I guarantee he has known it for over a year, as babies learn to associate the word (their name) with themselves by their 1st birthday. Now that I see the responses, I may think twice about nicknaming my wonderful babies in the future.
I guess since it bothers you so much, talk calmly to your daycare provider, and a good daycare will be completely understanding and strives to work with its parents. But I still think you should keep in mind it's likely a phase like most things 2 year olds do.
I have another question though, does he respond to his nickname? I ask because if he responds to SOMETHING, then it is a phase, if not, you may want to watch him closely for other developmental/behavioral issues such as him not looking at your face, little-to-no eye contact, speech or social problems. A child with late onset Autistic-spectrum will really start shows signs of it around 2-3. This was also the age my son really starting signs of it as well.