Not exactly, from what I can tell. This lets you run an individual process in a host environment, but using fine-grained privilege restrictions, and some judicially inserted virtualization. LXC runs hosts in a complete OS-level virtualized environment, more like its own Linux instance, which is a bit more complete virtualization (not always desired for a single process).

A Solaris / OpenSolaris / Illumos analogy is to compare running a process in a new Zone (which is like a FreeBSD Jail or a Linux LXC/OpenVZ container), versus running a process in an existing Zone but using ppriv(1) to selectively drop privileges usually given to processes by default, such as network access, file read/write, exec, etc. One is in what looks like its own OS instance, while the other is in the parent OS instance but sandboxed. The analogy isn't quite complete because the ppriv(1) approach doesn't lie to processes that try to violate the permissions like Mbox does; instead it just denies their attempts to do something that violates policy (such as opening a file). So, it might cause sandboxed processes to crash, rather than letting them complete with faked completion.

Not really; mbox uses seccomp-bpf to filter system calls, from userland, using unprivileged users. A very ambitious project to build on top of mbox might be an even lighter-weight Docker-alike using userland system call interposition instead of Linux containers.