Dragons are awesome so it makes sense that they appear all throughout video games.
They're so versatile. They can be feared monsters or wise ancient beings that can tell you what you need to do next. They can be cute or intimidating. They can be big or small.
Need something cool to base your boss on? Make a dragon. Need some legendary creature to be feared throughout your fantasy world? Make a dragon. Need a cool looking playable character? Make a dragon.
It's no wonder they appear all throughout the medium. And there's not just one kind of the iconic monster that you can base your creatures off of as well. It seems like every culture around the world created their own dragon species each of which have their own qualities that can be translated well to a video game character.
For example, dragons from Egyptian, Chinese, Greek, Roman, English, and Japanese mythology are so different that they might as well be separate species, and they're all perfect to be put into a video game.
That or you can make an original dragon, just give something the ability to breathe some kind of element and wings and it qualifies.
You know a game called Dragon's Dogma is going to contain dragons in it, if a video game has the word "dragon" in it, you kind of know what to expect. Although, Yakuza: Like A Dragon doesn't really have any in it so I guess that rule doesn't always apply.
Anyway, in Dragon's Dogma, your life gets kind of ruined by the dragon Grigori when he attacks the town of Cassardis, home to you and all your Arisen buddies.
After you do some general questing the bugger comes back and kidnaps your love interest so you have to climb the Tainted Mountain to rescue them and slay Grigori. Who he kidnaps is up to you as almost any character can become your love interest, it's a pretty cool system.
Anyway, Grigori is a great dragon, the fight against him is intense and a lot of fun, Dragon's Dogma has its problems as a game but Grigori isn't one of them.