Exactly, that's the idea. Most people do in fact request an invoice, except for very small purchases (like an expresso). In some areas where tax evasion is still rampant (like construction), they won't ask for an invoice because the contractor will give them a VAT-free price. On their side, the advantage is that they won't need to report the amount as income.

In addition to the lottery, there are also tax rebates for people who ask for invoices. For restaurants, I think it's like 20% of the VAT of the purchase, towards a limit of 250 EUR/year. Other industries where tax evasion is endemic (hairdressers, auto repair, etc.) have separate rebates and limits. I think this is the reason, more than the lottery, that makes people ask for invoices.