refactor: Allow specifying name of template loggers

src/mkdocstrings/handlers/base.py

@@ -74,6 +74,7 @@ class BaseHandler:
     To add custom CSS, add an `extra_css` variable or create an 'style.css' file beside the templates.
     """
 
+    # TODO: Make name mandatory?
     name: str = ""
     """The handler's name, for example "python"."""
     domain: str = "default"
@@ -132,7 +133,7 @@ def __init__(self, handler: str, theme: str, custom_templates: str | None = None
             auto_reload=False,  # Editing a template in the middle of a build is not useful.
         )
         self.env.filters["any"] = do_any
-        self.env.globals["log"] = get_template_logger()
+        self.env.globals["log"] = get_template_logger(self.name)
 
         self._headings: list[Element] = []
         self._md: Markdown = None  # type: ignore[assignment]  # To be populated in `update_env`.

src/mkdocstrings/loggers.py

@@ -177,10 +177,14 @@ def get_logger(name: str) -> LoggerAdapter:
     return LoggerAdapter(name.split(".", 1)[0], logger)
 
 
-def get_template_logger() -> TemplateLogger:
+def get_template_logger(handler_name: str | None = None) -> TemplateLogger:
     """Return a logger usable in templates.
 
+    Parameters:
+        handler_name: The name of the handler.
+
     Returns:
         A template logger.
     """
-    return TemplateLogger(get_logger("mkdocstrings.templates"))
+    handler_name = handler_name or "base"
+    return TemplateLogger(get_logger(f"mkdocstrings_handlers.{handler_name}.templates"))