While most bears will leave the area when people are around, young, old, starving, or hurt bears are the ones that will most likely stalk and attack a human for food, so I don’t dismiss a bear “stalking” a person. Sows with cubs will retreat when possible or clearly let you know you’re too close.
The part that makes me think it’s not a bear is that they don’t close in and attack. Bears are not overly smart critters. If they think they want you, you’ll see them.
As far as something different altogether, let’s just say I’ve heard a LOT of different people describe their “stalking encounters” in the woods...and they have similarities to your story. Some saw things, others didn’t . But the encounter characteristics often included keeping step with the person, stopping when the person stopped, resuming when person resumed, breaking branches or banging on trees.
PM me if you are interested to hear people tell their accounts.