The first response was not very clear. Of course you can sync phone with a computer running OS X 10.6.8, you just can't sync a phone running iOS 9.
iOS9 on a mobile device requires iTunes 12.3 or higher which in turn requires a computer running OSX 10.8.5 or higher. System Update only checks for updates for the current system version you are running but that itself may be out of date. It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system you will not be able to sync this phone to your present computer. If it can run 10.8.5 or higher you can either buy a download code for OSX 10.8 Mountain Lion from Apple online, or you can try installing the free El Capitan OSX 10.11. El Capitan may run slower on older machines and require additional RAM purchase. Doing a big jump in system versions is also more likely to affect old software.
Snow leopard to el capitan, will it cause my macbook to be slow? - https://discussions.apple.com/thread/7412959
Mountain Lion 10.8 purchase link U.S.A. - http://www.apple.com/shop/product/D6377Z/A/os-x-mountain-lion
Mountain Lion 10.8 purchase link U.K. - http://www.apple.com/uk/shop/product/D6377ZM/A/os-x-mountain-lion
Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/
If your wife's computer is indeed nine years old you will not be able to upgrade to the required OS X. There are workarounds to enable her to use the phone in a limited capacity using iCloud.
If any part of your home technology is older than five years it is critical you read the fine print on any equipment you purchase. It is standard for Apple devices to require the current version of iTunes at the time of purchase of the device. Recently Apple has been doing annual software upgrades which is why the requirements for your device that you purchased a year ago and the requirements for your wife's device are very different.