The streaming messages are part of a binary log file. Apple are moving to a unified logging system, so the logs do not work the same way anymore, at least by default. One might try to configure syslogd and/or ASL (Apple System Logger) via one or both of the configuration files, /etc/syslog.conf and /etc/asl.conf. (There are many more configuration files for ASL stored in /etc/asl.) This might involve grokking manual pages to learn how to produce plain text log files for certain messages. Some of these manuals might be man 1 log, man 1 syslog, man 5 syslog.conf, and man 8 syslog. I have not yet determined exactly how to do this, but I wanted to give you some sort of method to produce, view, and "tail" custom messages. (I'll come back and update this answer if I ever do learn how to send certain messages to a plain text file by configuring their new logging system. It seems that it, at least, can be done, because there do exist plain text files in /var/log.)
One can send messages with syslog -s and logger. (The -s option sends log messages to syslogd, and syslog can do more than logger.) An easy way to follow the logs in a similar way to tail is to utilize Console.app. Open the program and place the cursor on the local machine listed under Devices. In the search field, provide the name of the program that was used to send the message: one of syslog or logger. Next, open a terminal and send a message.
syslog -s "test message"
The message can then be seen in the Console as illustrated below.

Messages can be acquired from the logs. Use the --info and/or --debug options to specify lines that log would ignore by default. Specify which process sent the syslog message within the --predicate option.
log show --info --debug --predicate "process == 'logger'"
or
log show --info --debug --predicate "process == 'syslog'"
To get an effect similar to tail, use the stream command.
log stream --info --debug --predicate "process == 'logger'"
or
log stream --info --debug --predicate "process == 'syslog'"
After starting the stream, send a message to witness that it does behave in a similar way to tail, as illustrated.

The command, syslog -s "test message" was used to send the messages.
logger xyzwill log to /var/log/syslog, but on macos, this file appears to be /var/log/system.log, but using logger doesn't seem to write to it by default