Hi SPCsteph,
welcome to the Apple (user-to-user) Community.
Re: "My second Apple Watch is paired to my phone but not showing up in the watch app"
Wondering which iPhone model, and iOS version you are using with the two watches.
The way I understand it (am no expert, yet your post surely deserves a reply!):
seems iPhones with iOS 16 or higher can pair with watches with WatchOS 9 or higher
Else:
Apple support has this article online, although it has been archived and is no longer updated:
Use more than one Apple Watch with your iPhone - Apple Support
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You may want to check Compatibility:
and also:
Update your Apple Watch - Apple Support
says:
"If your Apple Watch has watchOS 6 or later, you can install updates without your iPhone":
Eg: Scroll to "Update directly on your Apple Watch" for "how to" / tips in the article above.
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Resource:
Apple Watch User Guide - Apple Support
For more support, info., contact, if needed:
Apple Watch - Official Apple Support
and
iPhone - Official Apple Support
All the best :-)