A product on your list is a few words. The page looks for notices in which every one of those words appears, in the brand, the title, the product names or the hazard text. It is a text match, so it can find notices about a different model of the same brand, and it can miss a notice that names the product differently. Treat a match as a reason to open the notice, not as a verdict.
A vehicle on your list is a make, a model and a model year. The page asks NHTSA for the campaigns filed for it, exactly like the vehicle lookup. The count tells you there is something to check; whether your own car is included depends on its VIN.
Nothing here runs in the background. The site has no account, no email and no notifications, so the check happens when you open the page. A bookmark and a look once a month is a reasonable habit: CPSC publishes most recalls on Thursdays, and openFDA is updated weekly.