I do this thing where I tap out syllables of words that I see or hear whenever I feel anxious. I can't stop doing it unless the anxiety completely goes away. Is this a compulsion or a skill my brain/body is doing automatically?

Do you feel like you NEED to do it to feel better? I could honestly see it being either a compulsion or a coping skill, like fidgeting, but tapping out syllables sounds more like a compulsion to me. If you feel like it's the ONLY way you can reduce your anxiety in the moment, I would look into it. I always encourage people to talk with a professional if they're worried about mental health concerns. This sounds like a good topic to explore with a therapist/psychologist. Think about any other thoughts/behaviors that could be obsessive or compulsive and see if there's a pattern that you can work on putting an end to. Wish you the best of luck!