In a very small number of cases, regional storage of data becomes regional restrictions. You see that already in how it's unlikely you will be able to access your consumer account from both Russia and the US at the same time unless you have a travel history to one or both countries. Similarly, an app storing data for EU citizens might indeed say, no, we can't share this data outside the EU, we're not allowed. How it figures out if you're outside a region is, of course, problematic.
But that's rare. Generally the data lives on a regional server and gets accessed by a CDN in a region near where the data will be delivered. If you're allowed to sign in from that region, then you're likely authorizing transfer of data (for the duration of that session) to that same region.
As for multi-region replication, it's often the case there are multiple regions in the same country - for example, us-east-1 and us-central-1. No jurisdictional problems to host the data in two regions in the same country, after all.
As to how you look up what region the user's data is stored in, well, you can keep a list of users and regions. Worst case, force the user to remember their region, and use separate copies of the website (and DNS) to pick a region to sign in on. For example, the same app/website, but at different country domains.