Hello
Typical solution for granting each Zshell a separate file is to use $$,
e.g.:

  LOGFILE=$HOME/.cache/zsh-logs/zsh.$$.log.

This means there will be enormous number of files created. The plugin
zdharma/zsh-unique-id allows to set:

  LOGFILE=$HOME/.cache/zsh-logs/zsh.${ZUID_CODENAME}.log

No two concurrent Zsh sessions will have the same code-name. $ZUID_ID can
be used to have numbers 1, 2, ... instead of code-names.

Could be an upstream autoload function a good idea for this?

  autoload -Uz acquire-unique-id
  acquire-unique-id  # sets ZSH_UID and ZSH_UID_CODENAME

The code is simple, it uses zsystem-flock (util-linux/flock for Zsh < 5.3,
but that doesn't matter for upstream), it's the second link. Plugin is at:

https://github.com/zdharma/zsh-unique-id

Business logic code:
https://github.com/zdharma/zsh-unique-id/blob/master/zsh-unique-id.plugin.zsh#L76-L156

-- 
Best regards,
Sebastian Gniazdowski