Make logging setup consistent between redbot and redbot-setup (#6695)

Co-authored-by: Michael Oliveira <34169552+Flame442@users.noreply.github.com>

redbot/setup.py

@@ -14,6 +14,7 @@ from typing import Dict, Any, Optional, Union
 
 import click
 
+import redbot.logging
 from redbot.core._cli import confirm
 from redbot.core.utils._internal_utils import (
     safe_delete,
@@ -436,15 +437,9 @@ def cli(
     overwrite_existing_instance: bool,
 ) -> None:
     """Create a new instance."""
+
     level = cli_level_to_log_level(debug)
-    base_logger = logging.getLogger("red")
-    base_logger.setLevel(level)
-    formatter = logging.Formatter(
-        "[{asctime}] [{levelname}] {name}: {message}", datefmt="%Y-%m-%d %H:%M:%S", style="{"
-    )
-    stdout_handler = logging.StreamHandler(sys.stdout)
-    stdout_handler.setFormatter(formatter)
-    base_logger.addHandler(stdout_handler)
+    redbot.logging.init_logging(level)
 
     if ctx.invoked_subcommand is None:
         basic_setup(