Hi KenTuckeyWoman,
Compatibility between an Apple Watch and iOS will be based on the version of watchOS installed. An Apple Watch Series 3 will support a maximum of watchOS 8 per watchOS 8 and watchOS 9 whereby watchOS 8 lists the Series 3 as supported by watchOS 9 indicates Series 4 is the oldest hardware capable.
The pairing instructions for watchOS 8 found at Set up and pair your Apple Watch with iPhone indicate "To use your Apple Watch Series 3 or later with watchOS 8, you need to pair your Apple Watch with an iPhone 6s or later with iOS 15 or later."
You should be able to complete this process without issue. Try fully erasing the Apple Watch if necessary and starting the pairing steps from the beginning.
Regards.