I am not sure if I am misunderstanding something fundamental here. My general approach is to have a connection per org. For  now these are all database connections.
Client App is configured for business users only:
    "organization_usage": "require",
    "organization_require_behavior": "post_login_prompt",
So I have an org
 {
   "id": "org_8Gewxxx",
   "name": "00f18930-xxx",
   "display_name": "White Region",
   "metadata": {
     "password_expiry_days": "365"
   }
 },
with a single enabled connection.
[
  {
    "connection_id": "con_XMjcxxx",
    "assign_membership_on_login": true,
    "is_signup_enabled": false,
    "show_as_button": true,
    "connection": {
      "name": "connection-for-white-region",
      "strategy": "auth0"
    }
  }
connection details
  {
    "id": "con_XMjcxxx",
    "options": {
      "mfa": {
        "active": true,
        "return_enroll_settings": true
      },
      "attributes": {
        "email": {
          "signup": {
            "status": "required",
            "verification": {
              "active": true
            }
          },
          "identifier": {
            "active": false
          },
          "profile_required": true,
          "verification_method": "link"
        },
        "username": {
          "signup": {
            "status": "required"
          },
          "identifier": {
            "active": true
          },
          "validation": {
            "max_length": 15,
            "min_length": 1,
            "allowed_types": {
              "email": false,
              "phone_number": false
            }
          },
          "profile_required": true
        }
      },
      "passwordPolicy": "good",
      "passkey_options": {
        "challenge_ui": "both",
        "local_enrollment_enabled": true,
        "progressive_enrollment_enabled": true
      },
      "password_history": {
        "size": 6,
        "enable": true
      },
      "strategy_version": 2,
      "password_dictionary": {
        "enable": true
      },
      "authentication_methods": {
        "passkey": {
          "enabled": false
        },
        "password": {
          "enabled": true
        }
      },
      "brute_force_protection": true,
      "password_no_personal_info": {
        "enable": true
      },
      "password_complexity_options": {
        "min_length": 12
      }
    },
    "strategy": "auth0",
    "name": "connection-for-white-region",
    "is_domain_connection": false,
    "realms": [
      "connection-for-white-region"
    ],
    "enabled_clients": [
      "Xoscxxx", //management api
      "ntW3xxx" //SPA client
    ]
  }
And a user with a single identity
  {
    "created_at": "2025-05-06T15:03:56.799Z",
    "email": "alan.hay+20@xxx",
    "email_verified": true,
    "family_name": "Hay",
    "given_name": "Alan",
    "identities": [
      {
        "connection": "connection-for-white-region",
        "user_id": "0955bf29-xxx",
        "provider": "auth0",
        "isSocial": false
      }
    ],
    "name": "Alan Hay",
    "nickname": "Alan",
    "updated_at": "2025-05-06T15:10:25.846Z",
    "user_id": "auth0|0955bf29-xxx",
    "user_metadata": {
      "check_email_verification": true
    },
    "username": "alhayxxx",
    "blocked": false,
    "last_password_reset": "2025-05-06T15:10:25.832Z",
  }
If would expect, if I am understanding the documentation correctly, that when the user belongs to only a single org then that should be resolved at authentication time and I do not need to pass the organization name or id with the auth request.
However when I do not explicitly send the organization id/name as part of the auth request then the login attempt is made against a connection to which the user has no relationship (but is enabled for the same client app).
So I am not sure if my understanding is fundamentally wrong or if some other config is breaking this. Or is this due to the fact that we use a connection per org rather than a single connection and at the point of authentication then without the organization param it cannot be determined which database to look in for the credentials? I am guessing this is my issue. If we were using email identifiers rather than user name then we could use HRD to determine the correct connection (assuming a single connection per org).
Log Extract
 {
    "date": "2025-05-07T09:12:18.920Z",
    "type": "fu",
    "description": "Wrong email or password.",
    "connection": "Blue-Region",
    "connection_id": "con_HfD7xxx",
    "client_id": "ntW3xxx",
    "client_name": "xxx",
    "ip": "xxx",
    "user_agent": "Chrome 136.0.0 / Linux 0.0.0",
    "details": {
      "error": {
        "message": "Wrong email or password."
      }
    },
    "user_id": "",
    "user_name": "alhayxxx",
    "strategy": "auth0",
    "strategy_type": "database",
    "$event_schema": {
      "version": "1.0.0"
    },
    "log_id": "90020250507091219005460000000000000001223372051404369952",
    "tenant_name": "dev-f2r5xxx",
    "_id": "90020250507091219005460000000000000001223372051404369952",
    "isMobile": false
  },