I got a bike for my birthday, a cheap one, 200€ 27,5 with front suspension, but I want a full-suspension bike. My question is: is it cheaper to buy a full-suspension frame and move the parts to the new frame, or buy a new bike?
You have a "BSO" and as such is built down to a price. A bare full-suspension frame will cost more than that whole bike, and no guarantee the parts will move over.
I suspect you're a teenager? Your best bet is to just ride your new bike. Develop endurance and leg power and aerobic capacity. While you're riding, look at other riders and learn.
Even though you call it a "cheap bike" its still worth looking after. Store it inside at night and not out in the rain. Tune it up occasionally, and don't biff it around. Lock it too.
In 3-5 years you can decide to replace it outright with something that you've saved up for - so budget to save 10 euros a week and in 3 years you'll have over 1500 euros to spend on the bike you want. By then your legs should be fully grown so the current bike frame will be too small.
And be grateful that someone bought you a bike, even if its not exactly what you wanted.