It's true. You're not. You're a person who might be feeling anxious right now. Or frequently. Perhaps more frequently than other people in your life.
But you are not an anxious person.
Emotional states come and go. Pain (physical and otherwise) is a transient state. In meditation, one of the reasons not to scratch an itch is to actually observe the itch. It's not stagnant: it might grow stronger, then weaker, then stronger again. It might burn a bit, or tingle. If you scratch it right away, you'll never know.
Playful curiosity of an itch might not seem worth it, but it's practice for what you feel off the meditation cushion: a craving, an emotion, or a physical pain. Everything ebbs and flows.
You can observe your anxiety the same way: with detached curiosity. Where do you feel it? A pit in your stomach? Tightening of the chest? Racing mind? Take a breath, and check in again. Did it change at all? Is it mixed with another emotion (anger, self-consciousness, fear)?
The difference between being a person with anxiety and identifying as an anxious person is significant. You are more than your emotions. You are also a person with skills to handle strong feelings. Calling yourself an anxious person amounts to a cop-out, to selling yourself short, and to giving yourself a life sentence as a person struggling with fear of that which has not yet occurred (the definition of anxiety). And you, most certainly, are a whole lot more interesting and intricate than that.