You have correctly surmised that iOS 15 is the end of the line for the iPhone 6s.
According to the App Store description, the developers of the Uber app have chosen to make it require iOS 16.2 or later, which means that you will not be able to run the Uber app on your iPhone 6s. That doesn't affect your ability to use other apps that are compatible with iOS 15 – but it does mean that you will need a new phone if you want to use that one.
A Help article on the Uber site indicates that the app works on iOS 8 and later. This is clearly an error on the Uber site (i.e., someone forgot to update the Help article to reflect the current app.)
https://help-stage.uber.com/riders/article/downloading-the-rider-app-ios?nodeId=aac76d4c-578b-4069-b0d4-0ef3ed792bea
I don't use Uber, but it might be possible to request a ride on your iPhone 6s by using Safari to visit the Uber site – even if the Web interface wasn't as polished as the dedicated iPhone app one.
https://www.uber.com/