Yes, they record the serial number . . .
I've taken my MacBook Pro to the Genius Bar several times in the past few months due to unusual, undiagnosed freezing. Each time I've brought it in and they've worked on it, they always check the serial number and enter it into their computer for reference. First off, they want to determine whether it is covered under original warranty, Apple Care, or not.
Similarly, each time I call Apple Care for support they ask for the serial number and make a record of the issues addressed and resolution (if any).
Finally, the last time I was at the Genius Bar, I was there for another reason, but asked some questions about my MacBook Pro. They were able to pull it up on their system to see what work had been done on it.
I believe the technicians at the Genius Bar will not and cannot perform work on any Mac product without accessing and recording the serial number.
A few years back, having trouble with my G3 ruby iMac, I naively carried it into the Apple Store (to various strange looks) and I was told they would not even look at it, they stopped supporting it and even if I was willing to pay for diagnosis and repair, they wouldn't touch it.
Do you have a reason to be concerned about their recording the serial number when they perform work on your unit? Wouldn't you prefer that they record that information?