You can do invoices with PayPal, however we don't sell anything with invoices because we don't deliver things with payment terms (future payments). An invoice would be if someone wanted a quote for something, but hasn't paid and then a company's accounts payable department mailed a check for it (an invoice is a request for payment).
A transaction receipt is more along the lines of what we do. Someone pays for something before they get it... similar to a grocery store or department store (a grocery store doesn't have an invoice for the buyer, they get a receipt showing the items purchased and transaciton info for the payment made). We aren't even collecting the info that would be needed to generate an invoice (for example company name, address, phone numbers, etc.), rather we are just selling specific items and generating a transaction receipt showing the item was paid for. Transaction receipts are printable from within your PayPal account if you look at the transaction detail (it has all the relevant info... what it was, when it was paid, the amount, the transaction ID, etc.)
An invoice wouldn't show something as being paid necessarily, it would only show the item and price and when the amount is due. If you need documentation that something was actually purchased, the transaction receipt is what you would want I would think?
If you look at a Stripe invoice, it doesn't show that anything was paid, it's an invoice, so it's just a request for payment. It shows only what is being sold and info for the customer on how they can then pay that invoice:
Basically there's the invoice (which is a bill/request for payment), and separately a transaction receipt which is the info showing something was actually paid. Since we aren't requesting payment, we don't utilize invoices within our business/accounting system since people pay first for the item(s).