So let me try to summarize the trends relevant to the OP's question before I throw a wrench in the works.

We have a definition of vehicle and operator which varies by state, which determines whether the state considers an infraction on a bicycle to be part of your overall vehicle or "motor vehicle" driving record.

We have a federal law that indicates you do NOT have to carry or produce ID (as opposed to a motor vehicle operator license when in a car) but do have to identify yourself. But absent some smoke & mirrors identity fraud (not advisable and an offense in itself), since the cops have computers and such, whether or not you present your vehicle license at the scene of the bicycle infraction is somewhat irrelevant. The event is either going to go or not go on your driving record depending on the state's policies if cited. And, of course, the cop's druthers to cite or not which may be influenced by your perceived level of cooperation, which may include the ID thing.

The level of information sharing between the state(s) and your insurance company may vary. There is some hope that they may not find out about a valid ticket, and some concern that they may find out about a bike ticket but incorrectly perceive it as a motor vehicle violation. Though I'm not sure if the latter means the insurance companies are expected to have their own bike-vs.-car policies that may vary from the states', or if that's simply a concern about the states' recordkeeping.

OK, now the trouble making:

I see this thread has rapidly trended to include a strong bill of rights theme about ID. I think there are a lot of problems with a "mandatory ID carry" law, primarily focused around the logistics of carrying while you're, for example, at the beach or out for a run. Ignore that, and just stick to the current requirement to identify yourself to the law enforcement officer in the event of an offense or probable cause.

==> Does anyone really have a problem with that?

Please do not answer based on some slippery slope. i.e. do NOT assume "identify yourself" is abused for fake stops, let's say you or the theoretical "they" really are an offender.