Can I ask, why would the car not come with plates?
In the US, a vehicle is purchased from the seller, then it is the responsibility of the buyer to register the vehicle with his/her local Department of Motor Vehicles (DMV) and secure a number plate for the vehicle after paying the requisite taxes and registration fees. In most states, the dealer will provide a temporary (paper) number plate which may be used for a period (usually 30 days) to allow time to secure the proper registration through DMV. In some cases, the seller (usually a dealer) will secure the number plates for the buyer, but this is a courtesy rather than a requirement. In the US a vehicle and its number plate are not wed forever. If a person sells a vehicle, they are usually required to remove the number plate once the sale is completed and the new owner must secure his or her own number plate.
Once a number plate is affixed to a vehicle, it must be renewed on a regular basis (usually yearly) by paying ongoing taxes and fees. There may also be a vehicle safety and emissions inspection required for renewal or required on a regular basis separate from number plate renewal. The OP was stopped because his rear number plate did not have the stickers indicating current payment of those renewal taxes and fees.
Insurance is required in all states at a certain minimum level and is a separate transaction from either purchase or registration with DMV.
If you are stopped by the police, you are generally required to produce a registration showing the ownership of the vehicle, proof of insurance for the vehicle (a separate document), and a driving license. You are not required to prove you (the driver) are the owner of the vehicle, but questions may arise if you aren't.
Some states require both front and rear number plates, some only rear plates. Many states allow the purchase of "vanity plates" which have numbers and letters that say something like "DRJFRMLA." Unlike the UK, these plates may not be sold to another person, but remain the property of DMV for use by the person who requested them.
This whole process varies from state to state. The US government is not involved in the motor vehicle registration process except to issue diplomatic number plates through the State Department (it would be your Foreign Secretary in the UK).
I hope that makes clear our process.