Yeah, alas all the ones listed are ones that I've already got on my little notes list of either have played, or can't access due to lack of the appropriate platform to play it on. But since it was asked, here's my list of dragon games with my personal descriptions/opinions:
Istaria: Chronicles of the Gifted - Probably the "old MMO" that Demonic Spoon mentioned. Actually one of my highest-rated games on this list, but absolutely not for everyone. Features lots of playable races, of which dragons (classical European style, quadrupeds with wings, fire-breathing, flight-capable) are one.
Has a lot of developments to make your dragons really feel like dragons -
Your armor class is heavily boosted by how much treasure you have in your hoard
You start as a hatchling, then grow to adult-size, and finally to ancient size, becoming the largest playable MMO race ever as far as I am aware, with a big jump in power for each stage as expected
You use a unique form of magic that no other race can access, Primal magic, which is considered the "source" of magic whereas other races' magic is derivative of Primal
You use unique equipment that no other races use
You have unique classes that no other races can join
You dig lairs down into the earth with unique structures and styles no other races use for housing.
Not specifically related to dragon gameplay, the game has a very detailed crafting system that is tied together with the adventure system but not dependent on it - you can get to level 100 in any particular crafting school (of which there are quite a few) without killing a single monster, although some resource areas are dangerous and you'd want a friend there to protect you.
The world is huge and mostly seamless, with only a few areas having dedicated "you have to load in and out of it" spots.
However, it is an old game, and it shows. It's very slow-paced and the level grind is a long one. All quests are your standard kill or fetch quests, and the combat is a sort of meandering pace that's too slow for many people. It's the kind of game you absolutely want to play with friends around to chat with and quest with, because by yourself it's just a long stretch of quiet as you run around to get things.
It does have a monthly subscription, although you can play for infinite free trial as a human character.
Dragon's Prophet - Another dragony MMO. It raised a pretty big stir when it came out, but the general tone of it was disappointment.
You play as a character who can communicate with dragons, and collect a Pokemon esque stable of various dragon types to carry you around and use in combat. When the game first hit the market using them in combat was basically worthless for everything below Legendary dragons, but they changed how the mechanics around it worked so you actually want to have them out with you as much as possible now. Unfortunately unless you pay cash or grind an incredibly stupid number of the daily quest reward tokens, you can only have... three? I believe it's three, dragons with you at one time--and for some dungeons you really want to be switching dragons rapidly to tank through some of the tougher bosses.
Combat's okay, but it's that style of active combat that leads to monsters feeling a little glitchy, because they'll teleport around small amounts to account for latency making you miss for no reason. It doesn't impact regular battles too much, but dungeon bosses can sometimes be a huge pain or outright impossible because they'll slip over into a wall and get stuck. Probably related to the same issues, the cutscenes tend to be wonky and not sync properly, as well as having no sound most of the time, but the story's dumb and predictable anyway so you're not missing much.
As an MMO to kill time and see pretty dragons it's passable, but it suffers from the problem of most F2P MMOs in that the cash shop does not feel optional at all with how many things you need out of it (bag expanders and dragon slot expanders especially), and the community is downright horrendous (expect every possible glitch and exploit to be used at your expense constantly, followed by a stream of people mashing "LMAOLMAOLMAO" in the chat that I kept permanently muted.) I've heard these two aspects are not as egregious in the EU release of the game, so I'm probably going to try that one out eventually and see if it's any more pleasant. The NA release is pretty much dead from what I've heard.
Dragon - A.k.a. "Dragon: The Game" or "Red Level Games' Dragon" since its simple title is a little vague. This one's currently in Early Access. It's an extremely early alpha right now with absolutely no finalized material; every single aspect of it is currently in a WIP state. Since the team behind it has no money currently (its EA release happened shortly after fiascos like The Stomping Land, and so has very few buyers-in) development is moving at a glacial pace, and it's impossible to say if it'll ever actually be finished. It's a very ambitious project, but right now there's no content worth talking about and the game doesn't actually have any systems or gameplay. It's entirely a "wait and see" case.
Dragons and Titans - Mentioned previously but I'm doing all my notes so why not. Same problem as every other MOBA; bad community, bad cash shop practices, repetitive gameplay that does everything in its power to remove you from any feeling of player agency or ability to actually be your character. Move around a chippiece that is your dragon but zoomed out too far to really see if it's actually a dragon or maybe just a paper airplane, try to kill the extremely rapidly-moving pixels of mobs and enemy players while being drowned out in a sea of slightly cheap particle effects. Left me with a bad taste in my mouth. You have to pay cash to access the single player campaign as well, despite its billing as F2P.
Divinity: Dragon Commander - Another one that was mentioned so I'll just give it a summary. Bit off far more than it could chew; jack of all trades, master of none. Not really that fun as either an RTS or a political sim, as both game modes just leave me wanting more options. Missing some oddly basic functions that limit playstyle to the single one they presumably used in their group testing, so don't try to get creative. You can't order more troops produced while in dragon form, so dragon form is pretty much only useful for the final mop-up at the end of the mad zerg rush that every single battle encounter turns into, since the enemy AI knows only a single tactic of "pump out the max number of units immediately and send them all towards you." The political segments just left me pulling my hair because you have absolutely no options to mediate or compromise between the various forces; you can side with one or the other and that's the end of it.
I of the Dragon - An old PC game recently re-released on Steam. Unfortunately, re-released without any bugfixing; the original release was put out early because of cut budget and development time problems, and was never patched because the producers went out of business shortly thereafter. The game is actually functionally incompletable in its current state; you can sometimes get past the particular progress-halting quest by luck, but it's never a sure thing, and if your luck is bad you're pretty much stuck for good (it's related to how the individual install chooses to interpret controls when you're stuck on the only land-based quest in the game, and since the game was never actually coded to work with land travel, it's entirely random whether or not it actually recognizes your inputs in a consistent enough manner to finish the quest.) It also has a ton of compatibility issues with later systems, from as minor as the sound going out to as major as trying to take down your OS when the game crashes.
As for the actual gameplay, it's repetitive but mildly satisfying. You have a good sense of power (most of the time), but the difficulty levels between dragons is wildly unbalanced. Try completing the "protect your town" mission as the Black Dragon when all the enemies that it spawns are immune to poison. Amusingly, in reference to the request about Necromancer games, the Black Dragon can actually summon the corpses of enemies back to life as short-term minions. Unfortunately, the spell is bugged--the corpse doesn't stand back up, but stays in its death pose, and so the attacks of most minions will clip into the ground and not work on anything, as well as them often being unable to move anywhere since their death pose has no movement code.
Scalebound has me excited and hopeful, but I don't own the console it's on, so. That's the end of that. No money to buy consoles.
There is a new Monster Hunter spinoff game coming that I can't remember the freakin' name of, in which you can raise at least some of the monsters including dragon and wyvern type ones (I think. I don't remember where to find the information and trying to google it is not really getting me anywhere right now for some reason.) Also has me excited and hopeful since I love the aesthetic of the beasts in Monster Hunter and am always sad that all you can do with them is kill them.
Dragons of Elanthia never went anywhere. It was going to be an arena based dogfighting aerial combat game. Now it's dust.
Time of Dragons is some facebook game where you ride dragons and pay lots of money into the cash shop so you can win.
Dragon Rage is a PS2 game where you play a dragon that uses various elemental weapons to defeat the evil Orcs. It's got wonky controls and bad missions, but it's somewhat fun anyway. The voice acting is atrocious but worth a laugh. "A dragon without gold is like a... dragon... without gold!" Your dragon will commit suicide if you fail certain missions, and that's alarming.
Those are just a grab-bag of ones that come to mind. There's more out there, but most of them are too similar to ones listed or are ancient flash games like Akai Ryu.