If I hover over a person's name, I get several choices, among them "Follow Member". This is so I can easily see new posts from wise and helpful members, like Speckled Hen, and I can tell when someone who's been out for a while, like dawg53 or patandchickens, posts again.
On the other hand, if I'm annoyed by someone's posts, I can choose "Block Member". Then I can resist the urge to say cutting things and make work for the mods
On the "old BYC," people could track your posts by searching for your username and then choosing to view all of your posts. "Follow" does make it a little easier to, well, "follow" someone, but it's not a new capability.