I'm using PuTTY default terminal emulator.

I configured PuTTY's Default Setting to Logging Everything Terminal Session.

like this. each opened session log file is identical. seconds, Host, Port level.

D:\junk\logs\&H&P-&Y&M&D-&T.log

GNS3 bundled PuTTY also load System wide installed PuTTY Configurations.

This behavior have issue that shows warning when open all terminal session same time.

---------------------------
PuTTY Log to File
---------------------------
The session log file "D:\junk\logs\192.168.229.129&P-20181129-081719.log" already exists.
You can overwrite it with a new session log,
append your session log to the end of it,
or disable session logging for this session.
Hit Yes to wipe the file, No to append to it,
or Cancel to disable logging.
---------------------------
Yes   No   Cancel   
---------------------------

I have closed this warning over, over, and over, and this means part of session do not logging because file name is conflicted.

Pretty much stressful.

I found GNS3 Build version Networking Lab useful options. -log override default logging configuration. And saved each terminal session logs appropriate file name like a 29.Nov.18_08.05.38_R1.log.

Default

putty.exe -telnet %h %p -wt "%d" -gns3 5 -skin 4

Specify log dir

putty.exe -telnet %h %p -wt "%d" -gns3 5 -skin 4 -log D:\junk\logs\GNS3

Happy.