Disclaimer: Not a doctor or nurse...
Yes. In a word, yes.
As the mother, sister, and daughter of asthmatics, I think you can assume it's asthma. It could also be other things, such as a heart murmur. (I have a leaky valve. It causes many of the same symptoms as asthma.) But, the chances of it being a congenital heart defect are not as likely as it being asthma.
I would tell the doc the same thing you told us. Let them know it gets worse after exercise.
Does it get worse at night, too? If so, tell them.
You should do a search for asthma. It can get you to some good resources.