Does it mean anything if you give a guy a New Year's gift and he says that he feels bad he didn't buy you anything?

Don't read into this too much yet. If you bought your girlfriend a gift and she didn't buy you one, she could feel embarrassed and feel badly they didn't think ahead of time and buy you a gift too. You were sweet to give him a gift and he does sound like a nice guy and I'm sure he appreciates it. Take it slow and easy and keep the gift giving at bay until you know him a little better. Marcy