The shopping basket concept is generally held to mean that the customer can buy multiple items in one go at a checkout.
However, most IATA-member airlines either do not permit or strongly discourage their agents from selling multiple, unrelated tickets in a single transaction. In some cases it is also not allowed by local law to sell certain itineraries.
This means that if you had a "basket" for several flights, when the customer took the basket to the checkout, it would have to be made clear to the customer that each ticket in the basket comprised a discrete and separate contract with the airline(s) involved, and that the fact that the flights were sold together in one sale did not mean they comprised a continuous itinerary.
Further the ticketing systems used by the airlines will expect a one-to-one match between the card transaction and the ticket, so you would possibly have to split out the credit card transactions to one per PNR.
In some cases it would violate local laws to offer certain itineraries that are physically possible to fly but are not allowed to be sold [for instance, on an otherwise domestic itinerary, flying between two points in the US via a stop in Canada, or flying Miami-Mexico City-Havana on a single ticket without approval from the State Department].
Separate to the legal issues, when a single transaction is done for multiple flights, it is understood that the flights are "on one ticket", which has numerous specific consequences. For instance if the first flight is late, the forwarding airline remains responsible for getting the customer to his final destination. In Europe, the forwarding airline is also liable for accommodating him overnight and providing food, even if the delay is not the fault of the airline. The luggage allowance of the passenger is determined by specific IATA or US DoT rules, rather than by the individual airlines' rules. For these reasons, airlines stipulate how their fares may be combined together to create a single ticket, and violating the airline's rules is usually technically difficult or at worse financially ruinous for the agent.
Therefore both consumers and the airlines expect it to be clear when flights are on one ticket and when they are not; the easiest way is to sell tickets separately [except to very sophisticated customers]. This makes having a single "basket" of flights a proposition that does not work out very well.