TLDR- Either will set you up fine for a career in aerospace.
There's rarely going to be a shortage of employment for good, qualified aerospace engineers (I'm an engineer and interviewer in the aero industry and I've seen plenty of vacancies around the enterprise), and I'd suggest going down the AE route if you can, as it's likely going to give you a bit more enjoyment and will lets you specialise more easily once you are into industry but ME is still going to let you into AE roles.
Below is an bit from a recent BAE Systems AE job ad....
"A degree in the field of Mathematics, Physics, Computer Science or Engineering (Mechanical, Electrical, Electronics, Aerospace, Systems or Software)"
Whichever you choose, best of luck and have a great time. Welcome to the profession 😀
Edit: Typos and extra info.