No, most states are fine with looking you up by yor numbers. I have gotten stopped before while walking and they will just look you up by the numbers if you don't have ID.
What you would need a physical ID for are for things like: going to the bar, returning an item without a receipt, buying alcohol, buying something restricted like a gun, opening a bank account, seeing an R Rated Movie, buying a rated M Game, getting a new iPhone on an agreement, or in some cases using a credit card. ---these are more than enough reasons why most people carry ID.
But police isn't one of them. They can look you up by the numbers or your SS#. But this is considered giving the officer a hard time, and they'll more than likely find more ways to get you in trouble as a result.