That caveat applies to actual remote filesystems like NFS or SMB.
If you're just using a local filesystem as persistent storage it should be fine. But, LMDB records process IDs in its reader table, and containers all number their processes starting from 1. So if you have multiple containers accessing the same DB at once, you're likely to get PID collisions and break reader consistency.
I run multiple docker containers concurrently accessing the same LMDB file in production without any issues. Indeed, it breaks when processes get the same PID, which can however be solved by adding the option 'pid: "host"' in docker-compose.yml for all containers using LMDB.