You're talking about two different places there. There's the part where it is saved. Then there's the part where is success page is outputted. The general "action" is located in register.php (if you haven't already noticed).
using all versions since 1.0 will store all that information in the database, inside the table called "user" ... the process is simple: once the visitor click to submit the form, the engine verify the information and store the data in the table "user", generate a new userid for that user