Thanks for your details notes, what you described matches what I've been experiencing too, and it's reassuring to know it wasn't something specific to my code/project. I'm currently using Unity 2021.3.16f1
and just published my game here https://paulz.itch.io/roundy-round.
It's definitely confusing/misleading that all iOS browsers are essentially Safari under the hood because they are required to use Apple's WebKit framework. I even tried disabling "Prevent Cross-Site Tracking" and "Block All Cookies" in iOS Safari settings but it made no difference (Godot web publishing has some notes about IndexedDB). I don't have an Android mobile device to compare. Interestingly even when I updated Chrome (on Windows) recently I noticed my game's IndexedDB persisted data seemed to get deleted.
I even considered using PlayFab or some server-based storage solution, but if you want to use "anonymous login" for convenience so your players don't need to create or remember an account name/password, I don't see how that's possible because you'd still need to persist a GUID in the browser somehow, without using IndexedDB I guess.