Why does man have its own user?

man (at least, the version implemented in man-db) can be set up with its own user so that its “cache” (which includes indexes, as used e.g. by whatis) can be owned and updated by that user (instead of root).

Additionally, man can be setuid to that user, allowing it to write to its cache directory without requiring specific permissions for users:

$ ls -ld /var/cache/man /usr/bin/man -rwsr-sr-x 1 man man 115600 Feb 19 2021 /usr/bin/man drwxr-xr-x 31 man man 4096 Nov 23 00:00 /var/cache/man 

The above shows that the man binary is setuid and setgid man, and man owns /var/cache/man.

In Debian derivatives this can be configured with sudo dpkg-reconfigure man-db. The explanation given there is

The man and mandb program can be installed with the set-user-id bit set, so that they will run with the permissions of the 'man' user. This allows ordinary users to benefit from the caching of preformatted manual pages ('cat pages'), which may aid performance on slower machines.

Cached man pages only work if you are using an 80-column terminal, to avoid one user causing cat pages to be saved at widths that would be inconvenient for other users. If you use a wide terminal, you can force man pages to be formatted to 80 columns anyway by setting MANWIDTH=80.

Enabling this feature may be a security risk, so it is disabled by default. If in doubt, you should leave it disabled.

“cat pages” aren’t useful nowadays, so configuring a system without support for them won’t cause any issues. It is also possible to support cat pages without allowing their creation by arbitrary users, and thus requiring a man user and setuid binary; various Unix systems used to be set up so that they would periodically generate all missing cat pages (see catman).

Even without a setuid man, a man user can be useful for man-related cron jobs (whatis indexes etc.).