Unfortunately, hydreigon are also like cats in a way. As fearsome, bestial pokémon, they take pride in their ability to hunt … but they also expect their young to do the same. Technically, there are actually two possibilities at work here, but both of them are very, very similar. Either your hydreigon is trying to show you his catch because he’s expecting praise for it, or he’s showing you his catch because he’s trying to teach you how to hunt. You’ll be able to figure out which by observing his body language when he brings you his kill. If he sits back and cranes his main neck or if he lies down and rolls over, craning his neck at the same time, then he’s searching for praise. (Both positions are signs of submission among the hydreigon.) However, if he’s pawing at the corpse and looking at you expectantly, then he’s trying to teach you that this is what a kill looks like.
In both cases, thank him for the offering, give him a treat, and dispose of the body (perhaps by burning it, if it’s large) whenever he’s not looking. If he’s trying to show you how to hunt, tell him politely that you appreciate his efforts but that it’s not necessary for you to hunt the way he’s trying to show you. You may even want to let him watch you cook and eat to show him that you’re perfectly fine without hunting. (Sometimes, this method remedies the other motivation behind displaying a kill, but usually, otherwise, that behavior will simply continue until your hydreigon realizes on his own that he doesn’t need to show off a kill to get affection out of you.)
As for your other pokémon, luckily, you don’t need to worry. It’s not so much because your hydreigon recognizes them as teammates as it is because your hydreigon recognizes them as tamed pokémon. Wild pokémon are much more of a challenge to catch and kill (as they more often run or congregate in large groups), and thus, your hydreigon will always go after them instead. Even when faced of a shortage of wild prey, hydreigon would rather go without killing anything at all than killing something owned by a trainer. I’m not sure how comforting that statement is, but at the very least, the point is that your pokémon are safe.