PCI and PCI Express devices are identified by Bus/Device/Function, which is necessarily unique per device in the system. Vendor and Device ID are just properties of a device found at a certain Bus/Device/Function.
When enumerating board, a driver will typically scan PCI configuration space (iterate through all the installed PCI devices), until it finds one or more devices that match the expected Vendor and Device ID, and possibly also the subsystem IDs. Once it finds a match, it should record the bus/device/function as a "handle" to the open device.
Properly written software should find all vendor/device matches, put them in a table, and let you pick which one you want to use (e.g. /dev/mydevice0, /dev/mydevice1, etc). However, I have seen lazy software that simply stops at the first match.