Well they didn’t say that they are specialist level at both. It could be that they are mid/good level at both, which isn’t a particularly egotistical view to have of oneself. I got the impression that they were highlighting their skillset is more broad than deep, which is basically what you’d expect from a good full-stack developer.

I’ve met a lot of programmers early in my career who were full of themselves, and a number of more experienced ones later in my career that understood their own capabilities very well.