This is known as the squawk or transponder code. You cannot enter class B airspace without one which is usually around major airports. However, this does not mean the radar cannot pick you up, it simply means your squawk code or ID is not next to the contact and therefore it would be considered and unidentified aerial contact. My brother in law is an ATC and I'm a hobbyist pilot. The tower often asks me if I desire radar service and I usually say no. My squawk then is set to 1200 to signify VFR flight. The tower radar still picks me up but the controller can opt to filter the contacts via some criteria (such as no ident or squawk). Radar at airports will always pick up the contact it just may not be able to distinguish what the contact is. If the radio in the aircraft is not capable of sending either squawk, altitude, or altitude and squawk then the contact is not distinguishable. But I guarantee you if I take my Cessna within 50 miles of Chicago O'Hare airspace they will pick me up and attempt to contact me via a COM frequency. If the radar at major airports did not actively scan this could present a safety risk since any small plane could just wander in unbeknownst to the controller. It is the job of the pilot to follow the FAR's regarding class B airspace transition but what if the pilot does not follow them? Under your scenario the controller would never know about the aircraft.
Not really. Airport radar may be capable of proper scanning, but under normal circumstances, it operates purely in transponder mode. Meaning that it sends out a very low-level signal and doesn't bother to check for echoes. Instead, every plane carries a radar transponder that not only tells the radar that something is there, but also the flight number, aircraft type, altitude, speed, and other stuff.