Yes, this is typically done in a commercial context, but it does not need to be exactly as described.
As you already noticed, developers should write unit tests. Those tests as you probably know only test the smallest distinguishable things in the code.
Higher up there should be integration tests - either as an extension to the unit tests or maybe in the form of testing the API. If the integration tests directly call the APIs, I would assume that they aren't written by a tester, but by a developer. If the tester is focused on testing the functionality, the APIs are not his main concern. If the tester has a more technical role, then they could be one of his responsibilities.
You as an (automated) tester write automation web tests via Selenium. That should be the highest level in terms of testing, as it tests all features combined. These tests will call a lot of APIs, but not through the APIs directly, but through the web interface. As that is another layer on top of the APIs, this is not the best place to specifically test the APIs.
The advantages of having three levels of automated testing is that you are able to test on each level specifically what functionality is added. The unit tests tests each function, the API tests test whether the APIs work as expected, and the Selenium tests are mimicking the behaviour of a real user. Without the API tests you are tempted to write more Selenium tests to cover the APIs as well, with the disadvantage that the Selenium tests typically require more maintenance and more time to run.
Another advantage is that API tests are more easily setup as performance tests: running the API tests on 100 machines with 100 calls per second is much easier than doing the same thing with Selenium tests.
Concluding: in my opinion having three levels of automated tests has a lot of advantages. Of course, it implies duplication of effort, but I do believe that it is worth the effort. It is also possible to have a more technical person (maybe even a developer) write the API tests, so that you as a tester already have a better feeling of the quality before you even look at it.