"Is it normal they don't give it back to me?"
Yes. Let me explain.
You can only get legitimate and genuine replacement parts from Apple and nowhere else. An authorised Apple Repair Centre, once a diagnosis has been made, have to order the part from Apple. They don't carry Apple stock, they're not allowed to. That's the rules. The part that's failed goes back to Apple and Apple send a replacement part once they receive it.
This whole process is called the Global Service Exchange Program or GSX for short. GSX ensures you as the customer gets a genuine and authentic replacement and the part that's failed is repaired and put back into the program. This helps - amongst other things - Apple keep an inventory on all the parts they make (so as to forestall forgeries and fakes), ensure and maintain quality and keep repair prices down for you, the customer.
". . . furthermore, is there a way to know they changed the logic board for sure?"
IMO the only definitive way to really tell is to strip the laptop down prior to sending it in for repair, make a note of the part numbers on the MLB and then compare them to the numbers on the replacement once you get it back. If it was me I would be content with the invoice. There may, possibly, be other ways, but these are the two uppermost in my mind.
Bear in mind you have a limited (90-day) warranty on the repair. So be careful not to spill anything over it when using it in future.
Good luck.
