Shared hosting sounds like you don't have your data stored privately and doesn't sound like less work for the company.

Don't look at the name from a technicians perspective, but from the perspective of a manager of a small startup who wants to reduce the overhead for hosting it's service as much as possible. Also serverless is not wrong per sé, it's exactly what you as the customer get.

You could spin it the same way for every other instance. Why do you call GoDaddy "shared hosting", in the end it's just a pod on a kubernetes cluster. So why don't you call it "private kubernetes pod"?