I cannot pair my iPhone 6s with an Apple Watch 3. It keeps saying I need to update software and my iPhone is up to date

If you have an iPhone 6s, you should be able to upgrade to iOS 13.

If you have an iPhone 6 or iPhone 6 Plus, then iOS 12 is the highest version you can have on these iPhones, and that will NOT work with the current shipping Apple Watch Series 3 running watchOS 6.

iPhone -> Settings -> General -> About -> Model Name

Not enough RAM in the iPhone 6 to run iOS 13. It was not arbitrary to not support the iPhone 6.