What's the purpose of the "bin" group?

The base-passwd package guarantees the presence of certain users and certain groups in a system. The documentation is in /usr/share/doc/base-passwd/users-and-groups.txt.gz (or /usr/share/doc/base-passwd/users-and-groups.html) (source available online in Debian Salsa):

bin

    HELP: No files on my system are owned by user or group bin. What good are
    they? Historically they were probably the owners of binaries in /bin? It is
    not mentioned in the FHS, Debian Policy, or the changelogs of base-passwd
    or base-files.

    LSB 1.3 lists bin as legacy, and says: "The 'bin' UID/GID is included for
    compatibility with legacy applications. New applications should no longer
    use the 'bin' UID/GID."

So, it's part of the LSB specification, and since Debian used to aim for LSB compliance, Ubuntu has a bin user and group.

/usr was the original path for user homes in historic Unix, so there was a bin user in charge of /usr/bin - compare how the root user is implemented in these (and also in some more modern ones - Solaris, etc.) systems, it has /, not /root, as home directory.