iWatch 2 compatibility with iPhone 5s?

Bought a used iWatch 2. It pairs fine with an iPhone SE, but when we try to pair with an iPhone 5s we get this error on the phone:

"iPhone out of date... you'll have to update it to the latest version of iOS."

<Cancel Pairing> <Update Now>

When we click "Update Now", it says "iOS 12.4.8. Your software is up to date."

I realize that Apple has dropped support for the 5s and this is probably the latest version of iOS that we can get on that phone.

However, several websites tell me that the iWatch 2 should work with the iPhone 5s.

Do we need to update the Watch OS on the watch? Or something else?

Thanks.