Current Workaround:
If you select the "Photon" engine on a Standard/Shared Cluster, they change the access rights of /databricks/native/proj-data and rasterio works fine.

The downside:
Pay for "Photon" compute to use a Python library, which do not use Spark/Photon engine -> $$$

Future:
I talked to Databricks support and they will check if the engineering team can activate the same for Standard engine with Standard/Shared access mode, because the folder content is definitely not sensitive.