Yes, they’re both created with clone, but with different levels of sharing. A pthread will share the virtual address space of its parent, which makes shared memory simple to implement; use the same pointer and you’re done. CoW is not “sharing” really, because you can’t communicate over it, it just saves some creation overhead.
With CoW, technically nothing gets copied initially, but as soon as the new process starts executing, it’s going to start copying the stack frame and any other regions it’s using. With a pthread you can be certain it will just copy the stack.
Context switches are usually cheaper when you don’t need to throw out the old virtual address space (and invalidate the Translation Lookaside Buffer). Pthreads share virtual address space, so there is no need to flush the TLB.
In a use case like Postgres, you don’t necessarily need to optimise for context switches. If you have a lot of concurrent connections, each of which has one process, then you’ll only hit limits with context switching overhead if very few of those connections are fighting over any locks or spending much time in IO at all. This is atypical, so usually those other factors hit you first.