mjhjmtu

/dev/log is the default entry for system logging. In the case of a sytemd implementation (this case) it's a symlink to whatever /run/systemd/journal/dev-log. It used to be a receiving end of a unix socket handled by syslog daemon.

~$ echo "hello" > /dev/log 
bash: /dev/log: No such device or address
~$ fuser /dev/log
~$ ls -la /dev/log 
lrwxrwxrwx 1 root root 28 Aug 23 07:13 /dev/log -> /run/systemd/journal/dev-log

What is the clarification of the error that pops when you try to write to it and why isn't there a process holding that file (output from fuser /dev/log empty?

The logging does work normally on the system.

~$ logger test
~$ journalctl --since=-1m
-- Logs begin at Thu 2018-05-24 04:23:46 CEST, end at Thu 2018-08-23 13:07:25 CEST. --
Aug 23 13:07:24 alan-N551JM alan[12962]: test

Extending with comment suggestions

~$ sudo fuser /dev/log 
/run/systemd/journal/dev-log: 1 311
~$ ls -lL /dev/log
srw-rw-rw- 1 root root 0 Aug 23 07:13 /dev/log

I am summarizing the comments to a complete answer. Note that @MarkPlotnick was the first to point toward the right solution.

As you can see in ls -lL output, the file pointed by you link is a socket, non a regular file or a pipe.

~$ ls -lL /dev/log
srw-rw-rw- 1 root root 0 Aug 23 07:13 /dev/log

Look at the first character of the output. That s means that the file is a socket.

You cannot use the redirection mechanism > of bash (or, AFIK, any other shell) to write in a socket because the shell will try to open the file and open does not support sockets. See man open for details.

You have to use a program that connects to a socket. See man connect for details.

As an example, you can use netcat or socat (see How can I communicate with a Unix domain socket via the shell on Debian Squeeze?).

For sake of completeness, you can use the redirection on pipes.

~$ mkfifo /tmp/fifo
~$ ls -l /tmp/fifo
prw-rw-rw- 1 root root 0 27 ago 15.04 /tmp/fifo
~$ echo "hello" > /tmp/fifo

Look at the first character of the ls output. That p means that the file is a pipe.

You should verify /run/systemd/journal/dev-log, its permissions and who's using it. You were checking a link.