Can the support site (or help desk) create tickets in multiple projects?

[seth_williams]

For example, the external customer-facing portal could have the "license type" create a ticket in a Licensing Project ... and the "bug type" create a ticket in an Product Issue Project.

[Nic Brough -Adaptavist-]

No, a Service Desk in Jira is a function of a single Project.

You'd have to do some clunky things with automations or scripting, which would be counter-intuitive for your customers.