It's possible. Honestly, the only way to know is to see a doctor and get some tests done.
Since shortness of breath can be caused by a lot of different things - heart issues, other lung/respiratory disorders, and yes, anxiety at times - you do need to be seen to find out for sure what you have.
Pain in your chest and shortness of breath are always, always reasons to go to the doctor. Even if it turns out it's being caused by anxiety (and make them do testing before they say that), you should know that so you can address your treatment for anxiety. Your mental health is as important as your physical health.
Let us know what happens, and I hope you get some good answers and relief soon.