How to identify to which queue a call belongs after is forwarded from an IVR

I need to clarify your question.

"Caller's calls are received by an IVR menu first, and based on the selected option, their call gets routed to the corresponding extension."

Do you mean the "corresponding call queue (extension)", right? E.g press 1 => route to "Queue 1"; press 2 => route to "Queue 2" etc.

And,

"Each extension has attached a different template message for each queue"

Do you mean "each call queue" has attached a different template message for SMS auto reply?

And,

"As I am handling calls for multiple clients ..."

Do you mean "multiple users" under the same RingCentral account or "multiple users" under different RingCentral accounts?

You did not mention which telephony session event filter do you use. But here is the trick to identify which call queue is having an incoming call (regardless of via an IVR or a direct call)

Provided that you subscribe for a company (account level) telephony session event notification

'/restapi/v1.0/account/~/telephony/sessions'

Then you would get these events after the call is routed to a call queue

{
    "event": "/restapi/v1.0/account/809646xxx/extension/62284877056/telephony/sessions",
    ...
    "body": {
      "sequence": 6,
      "telephonySessionId": "s-a0d17e304ee1ez18a1e628a5fzcdfd0840000",
      ...
      "parties": [
        {
          "accountId": "809646xxx",
          "extensionId": "62284877056",
          "id": "p-a0d17e304ee1ez18a1e628a5fzcdfd0840000-3",
          "direction": "Inbound",
          "to": {
            "phoneNumber": "+1209788xxxx",
            "name": "Paco IVR",
            "extensionId": "62284877056"
          },
          "from": {
            "phoneNumber": "+1650224xxxx",
            "name": "WIRELESS CALLER"
          },
          "status": {
            "code": "Setup",
            "rcc": false
          },
          ...
        }
      ],
      "origin": {
        "type": "Call"
      }
    }
}
  
{
    "event": "/restapi/v1.0/account/809646xxx/extension/62284877056/telephony/sessions",
    ...
    "body": {
      "sequence": 7,
      "telephonySessionId": "s-a0d17e304ee1ez18a1e628a5fzcdfd0840000",
      ...
      "parties": [
        {
          "accountId": "809646xxx",
          "extensionId": "62284877056",
          "id": "p-a0d17e304ee1ez18a1e628a5fzcdfd0840000-3",
          "direction": "Inbound",
          "to": {
            "phoneNumber": "+1209788XXXX",
            "name": "Paco IVR",
            "extensionId": "62284877056"
          },
          "from": {
            "phoneNumber": "+1650224XXX",
            "name": "Demo Queue - WIRELESS CALLER"
          },
          "status": {
            "code": "Proceeding",
            "rcc": false
          },
          ...
        }
      ],
      "origin": {
        "type": "Call"
      }
    }
}

The extension id in the parties array is the extension id of the call queue that receiving the incoming call. The from.phoneNumber is the number of the caller. So, map the call queue extension id to the message template and you can get the right template for each call queue and reply to the from.phoneNumber.