The Tor Browser Sandbox is pretty great (thanks Yawning!) but how can one run multiple instances of it at the same time?
This isn't really something that's supported, because it's an uncommon use case and the code is still at the point of "it should do common things really well".
That said, if you set
$XDG_DATA_HOME to unique directories that exist (with appropriate permissions in the case of the runtime directory), it should work.
The use of the various directories is per the XDG Base Directory specification, available at https://specifications.freedesktop.org/basedir-spec/basedir-spec-latest.html