Testing an HTTP API involves running the server, and running a client against that.

Postman, insomnia, and Milkman are REST clients.

I haven't tried it, but I think it'd be quite awkward to get a server setup from Postman. I think it would at least involve calls to external programs to start a server, or to run against an already running server.

The more apples-to-apples comparison would be using Postman vs the chai tests. I'd argue that writing tests in JavaScript with something like Chai as described in the post is better than postman, because the code is in 'readable plaintext' rather than JSON.