I know this thread is over a week old, but I had to add a comment to this debate.
I believe without a doubt that bettas have emotions. Including anxiety in many forms.
My betta used to get mad at me after destroying his bubble nest every week during tank cleaning. For a day or two after destroying his masterpiece he'd flare at me when I would go to his tank as if to say "Don't you DARE go near my nest!" After a few days he would forgive me, stop flaring, and build a new nest. This went on for several weeks. I've had him for 3 months now and I'm pretty sure he is used to it now although he doesn't like it. I would put money on the thought that he actually knows when I'm going to clean his tank. He will look up at his bubble nest and kinda slump his doral fin. Looking sad. Fortunately, he is generally a happy fish. He gets attention from my son and my dogs. He loves to be watched. He displays his fins very large and swims from side to side in his tank for anyone that wants to see. :) He's felt mad, sad and happy why would he not feel anxiety from separation? I love him, he knows it because of the way I care for him, and in turn he loves me back.
My betta (Finley) definitely knows who I am. He has different behaviors to me in comparison to others.
Bettas are very smart and perceptive.
That is all! LOL