I am not aware of any law that requires a US citizen to carry identification at all times. I would not think that such a thing would be required of a cyclist since no license is required for cycling.

I do not think that it is a good idea to ride without ID. I know a lady who had an accident and ended up in a coma in the hospital and no one knew who she was. I also know of a guy who had an accident and ended up in a coma and later died. Having ID makes it easier on your next of kin.

As for the police, if you are stopped, it is nice to have ID. You are not required to have any. But if you don't have any and refuse to identify yourself, you might just get hauled down to the local police station or jail. The police have ways of dealing with uncooperative people. Sure you can get off later. But do you really want to enjoy the hospitality of the county?