To answer your basic question, yes, each user gets his own set of global values. They are not preserved once the user logs out. To preserve a user's values, the normal method is a Users or Sessions table. This is done by account name and allows you to store each user's values upon logout.
Now, with regard to a serial number, you can do what you're asking using a common table with a single record. However, be careful to trap errors in case more than one user attempts to access the record at the same time. You can wind up with a mess if that happens. Another method would be to record each user's value in his Users record and then use the maximum value entered during that day, updating it with a batch script at night. Your use of FileMaker's auto-enter serial capabilities to manage such a system may, in fact, be the best option (due to its intrinsically multi-user safe nature).
Here's a whitepaper for folks who are new to Filemaker. It's been a few years since I've read it but there is some good stuff. The very first topic is global fields.