You generally cannot do that with a "normal" syslogd because it is against the design of syslog facilities. Windows log forwarders work that way because Windows log events do not have a Unix-like facility and assigning one for the whole server is the easiest solution.
On Unix every log message has its own facility property to distinguish between different services on the same host. Later on even a central log server can identify the daemons with a filter on the combination of host and facility.
You plan would require a syslog forwarder to rewrite the facility. That is certainly possible, e.g. syslog-ng has rewrite rules and larger tools like logstash should support it as well -- but I would advise against that.
You would end up with your own site-local design, working against common practices, making every future change to the logging system more difficult.