This practice is entirely compatible with the PostgreSQL License, but it is often prohibited by GPL variants. You typically cannot combine GPL code with code under most other copyleft licenses, such as the Eclipse Public License.
Regarding copyright status, AI-assisted work is increasingly recognized as copyrightable in many jurisdictions, provided the process involves a sufficient level of human creative input (though the specific threshold varies by jurisdiction). Only work generated purely by AI, with no human involvement, is arguably public domain. In a case like this, which is akin to "pair programming," the output is almost certainly copyrightable.