Lego logs to stderr. To capture logs through tee, you need to redirect that output:

$ lego ... 2>&1 | tee -a lego.log

I would like to see the standard behaviour, that errors go to "stderr" and info and related to stdout.

You can take advantage to cron with filtered stdout but enabled errors to receive an email with the error message when a error is logged on stderr.
Or any other ideas, how "email me when error" could be done with lego (without a huge wrapper script) ?

My knowledge in go is limited - but it seems new logger instance is needed?

I support this. Errors are supposed to go to the stderr and stdout is for the rest.

for example - this message goes to stderr, but it is expected behavior.

2022/05/26 21:41:37 [domain.tld] The certificate expires in 89 days, the number of days defined to perform the renewal is 30: no renewal.

Are there any chances to get it implemented?