Looks like they did not include the creation of the counter in the program but instead are handling it on the Client side using System Program. Take a look at the tests: 

https://github.com/solana-developers/program-examples/blob/d6cd69d8df3eb95966b3d2361ede84529413567a/basics/counter/native/tests/counter.test.ts#L37C8-L43

This is a little different than how they've implemented in the anchor example, where they've explicitly created an initialize function in the program:
https://github.com/solana-developers/program-examples/blob/d6cd69d8df3eb95966b3d2361ede84529413567a/basics/counter/anchor/programs/counter_anchor/src/lib.rs#L11-L13