When using a client pool with a uri that has the ?appName=<something> option specified, none of the clients received from the pool will have the name passed to it. This is caused by the following lines 752-757 of mongoc-client.c, duplicated here:

   appname =
      mongoc_uri_get_option_as_utf8 (client->uri, MONGOC_URI_APPNAME, NULL);
   if (appname && client->topology->single_threaded) {
      /* the appname should have already been validated */
      BSON_ASSERT (mongoc_client_set_appname (client, appname));
   }

single_threaded is set to false when using a pool, as set at mongoc-client-pool.c:111.

This seems like a bug to me, but I could be misunderstanding how pools are meant to interact with the appName. The setting is explicitly excluded for multi-threaded contexts, so maybe this is intended. If it is indeed a bug, it should be fixed.