This is a misnomer. Saying we have a similar scope of practice would be like saying AAs have a similar scope of practice to whom they are assisting, their physician anesthesiologist. They are technically allowed to perform all aspects of anesthesia as long as an anesthesiologist tells them and directs them while they do it. A CRNA is licensed to perform all aspects of anesthesia, without being directed or supervised to do so. As I said in a recent post people get confused with CMS billing regulations asking in about 33 states for a physician (any physician/dentist/podiatrist) to sign the chart in order to get paid. This is not them directing the CRNA or even supervising their practice, and again this isn't tied to their license to practice, just getting paid by CMS. This is not true for AAs who's actual license in the 17 states that allow them to practice require specifically a physician anesthesiologist to direct and supervise their practice at all times.
Where some people get confused is when a CRNA chooses to work in an anesthesia practice with AA's that requires strict medical direction. This was their choice, people have different reasons for why this was necessary for them at this point in their life. While they work there they will have to conform to a practice designed to cover AAs scope of practice, essentially reducing the CRNA to half capacity. At any point, they could simply leave and go open up their own CRNA only practice if they felt inclined to do so. That is the real difference between them.
If you get into CRNA school you'll begin to understand the vast difference in training, mentality, and scope of practice and independence between an AA and a CRNA.