I'm not a controller, but I am very familiar with the radar systems they use.
When you request flight following, the controller generates an abbreviated VFR flight plan for your callsign. This is not the same flight plan that you file with Flight Service.
There are two types of abbreviated flight plans: One is "local," and the other is "NAS," a national airspace system flight plan. In probably 90% of cases, the squawk code you get will tie your callsign to a local flight plan.
The computer entry for a local plan requires only a callsign. Aircraft type and destination can also be added. The entry is very simple: N1234 C172 JFK <enter>. The radar display nearly instantaneously shows a code that the controller tells you to squawk. When you squawk the code, the radar system "tags" you, or places your callsign and information the controller entered about you next to the target squawking your code.
There are several code blocks assigned to each facility, which are used for different purposes. From your last post, I know you fly near Portland, ME. The Portland TRACON is assigned the following code blocks:
Facility management decides what to do with each of those code blocks. Normally, it's (1) IFR, (2) VFR Flight Following, (3) VFR Practice Approach, and (4) other. I don't know the specifics of their code configuration, but those are what the facility has been assigned to use (FAA 7110.66D JE SUP 1).
There are basically three radar systems in use right now: Common ARTS, STARS, and HOST/DSR. Portland has ARTS-IIE, which is a Common ARTS system. They can pass local flight plans to other Common ARTS TRACONs, but only your callsign can be passed along. When you are switched to a neighboring approach control, Portland may say "Advise ABC Approach of your type aircraft and destination." Note that Common ARTS cannot pass along any local plan information to a STARS-based approach control, or a Center, which is based on HOST/DSR.
Controllers on Common ARTS have to go to a separate computer to enter you into a NAS abbreviated flight plan. This takes more time and is rarely done, but the benefit is that all of your information -- aircraft ID, type, and destination -- can be passed to any radar approach control or center.
Controllers using STARS will usually enter you into a local flight plan initially, but the STARS system permits controllers to change your local flight plan to a NAS flight plan with a simple keyboard entry. They'll normally do this in order to hand you off to a center or a Common ARTS-based approach control. Finally, if you begin flight following with a center, you are automatically entered into the NAS because their computer system only generates NAS flight plans.