Is it possible to specify custom error log format on Nginx?

You can't specify your own format, but in nginx build-in several level's of error_log-ing.

Syntax: error_log file [ debug | info | notice | warn | error | crit ]

Default: ${prefix}/logs/error.log

Specifies the file where server (and fastcgi) errors are logged.

Default values for the error level:

1. in the main section - error
2. in the HTTP section - crit
3. in the server section - crit

In my error_log, time always presented int begin of each error string in log.

There is a hack for that.

We know that we can customize the access log format but not error log format. So the hack is, for customized error log, we generate access log only when error occurs.

This can be done using error_page directive.

http {
...
  log_format custom_combined "...";
  server {
    ...
    error_page 50x @create_custom_error50x;
    ...
    location @create_custom_error50x {
      access_log path custom_combined;
      return 50x;
    }
  }
}