In my understanding (working in advisory support for Azure), no, both services currently store the data as LRS. I believe replication across regions is a roadmap goal for some point in the future.
Here's the one link that mentions this:
Data is stored encrypted at rest in Azure storage, to ensure data
confidentiality, and the data is replicated within the local region
using locally redundant storage (LRS).
https://learn.microsoft.com/en-us/azure/azure-monitor/platform/data-security#3-the-log-analytics-service-receives-and-processes-data
The only current workaround I am aware of is that the Windows Log Analytics agent allows you to multi-home (send data to two or more workspaces). If you wish to hedge against regional outages, you can ingest the data in another region, but you will pay full costs for ingestion etc. in that region.
Multi-homing for Linux is still in the works as far as I know.
https://feedback.azure.com/forums/267889-azure-monitor-log-analytics/suggestions/10852986-multihome-agents
I'm not aware of any workaround for App Insights.