Hii,

No, AME CET is not fake, it is a genuine exam conducted to do course in aviation as Aircraft Maintenance Engineer(AME). It is a professional course as to do this course candidates must clear the exam.  The institutes available for the candidates are approved through the Directorate General of Civil Aviation (DGCA), Government of India.

Yes, you can pursue Aerospace engineering through AME CET.

Hope this helps.
All the best!!