If you are using the "private browsing" feature of the newer versions of Safari, and if the site is logging you based on your IP address + cookie, then you might well be able to log more unique hits than otherwise. It's really hard to say what is going on because the developer of the site is in control of the algorithm that decides what is "unique". If your IP address changes between sleep/wake, the change in IP is what is causing it to be logged. Usually, an IP address is not considered unique in itself because people behind a NAT router may all be identified by the same IP address, so a combo with some other parameter is used to further "uniqueify" the user.
Apart from "private browsing" I can't think of any reason their logger would perform so miserably aside from poor programming or perhaps a reliance on some ActiveX control that isn't available on Macs.