No it isn't. Maybe it's still installed on most shared web hosts (no idea, I haven't used one in years).
It isn't installed on virtually every linux server. It remains somewhat common, and if you're using a shared host you may not easily be able to use an alternative. In that situation, which I suspect to have been a minority situation for some years, it still makes sense to use mysql. For most of us, it makes sense to type the single line it takes for our package manager to install our database of choice.
Anyway, the point I was making is that mysql is widely available and highly stable.
So is postgres. In general there's no reason to recommend mysql, although it may be the right choice in some niches.