Deep Links on iOS should open the app if it is installed or take the user to the App Store to download it.
You may be testing one or both scenarios from a webpage or public link. If this breaks its typically a change in iOS major/minor versions, but this can fail for a wide range of reasons.
This is technically different from Apple's Universal Links introduced in iOS 9. But while googling you may see overlap in the term Deep Link & Universal Link.
Here are some resources to help troubleshoot the problem:
Official Apple guide to troubleshooting Deep Links: https://developer.apple.com/library/archive/qa/qa1916/_index.html
Simple options to assert the Deep Link is working https://blog.branch.io/3-steps-to-troubleshoot-ios-9-and-ios-10-universal-links/
This article from 2017 is a fairly good source and has a nice visual breakdown https://abhimuralidharan.medium.com/universal-links-in-ios-79c4ee038272.