airlines refuse to send invoices for flights booked through their official website [closed]

In the past, I booked air tickets through travel agency, and they sent both paper receipts and invoices through mail with no problems.

Recently, I started booking air tickets through specific airlines' official websites because it's more convenient in terms of searching for flights with desirable departure/arrival times and methods of payment.

However, when I called the airlines CS to send me both the receipts and invoices for the booked flights (before I used the ticket), CS said they could send an electronic receipt but could not send an invoice. I asked for the invoices because the finance office in my institute prefers invoice to receipt. This happened with Cathay Pacific and Singapore Airlines. Not sure about other airlines.

Have anyone encountered similar situations? Any idea why? Why are travel agencies able to send invoice while (some) airlines are not?

Clarification 1: Judging from the comments and the one answer so far, it seems that "receipts" and "invoices" mean different things in different settings (and possibly in different parts of the world).

Here I elaborate on the practice of the mentioned travel agent. Hope this makes the question more clear. After I paid the travel agent for the air ticket and before the trip, the agent would send me, by local mail, two documents, in ONE envelope. One was titled "receipt", and the other was titled "invoice". On the receipt, a simple sum of money that I paid was shown along with a short description of the itinerary. On the invoice, a detailed breakdown (airfare, tax, fee, etc.) of the sum of money paid was shown, along with the itinerary.