No, usually it does not.
Government registering of financial institutions usually is to make the government safe (eg FINTRAC is watching for money laundering and financing terrorism) rather than to make it's customers safe.
Most governments have many levels of registrations and regulatory bodies. The most stringent requirements are usually obligatory only for banks, and they indeed often include precautions for insuring customer's deposits. Even this insurances have limits, eg in most EU countries the state guarantees deposits up to 100kEUR. If you deposit more and the bank flops - you lose everything over the limit.
Companies like forex or currency exchanges usually make their best effort to avoid as many regulations as possible, just because it's costly. If a given company does have guarantee funds and/or customer insurance, it should be advertised and explained on their website.
However the whole issue of trust is misguiding. You don't have to "trust" in your grocery store to shop there. There is no government guarantee that the vegetables sold will be tasty. If you buy and the product fells short of your expectations, you call it a loss and start shopping elsewhere. Financial services are no different than any other product. I recommend to your aunt to start small and see how it works. If a service turns out well, she can increase the amount sent through exchange and decrease amount sent through bank. But still, it's always prudent to send eg $1000 every week instead of $4000 once a month. It's more time consuming and cumbersome than having your bank do it - but it's the safety and convenience you're paying premium for.