The first of these characteristics that I notice in all RPGs is an overall sense of progression. By that, I mean that as the game goes on, there is generally a sense that the protagonist is growing and getting better at certain feats. Most of these games accomplish this through an experience/leveling system. As players accomplish objectives and dispatch enemies, they gain experience. After enough experience, they level up and gain stats and/or skill points used to purchase abilities. This model is one of the most common, appearing in Final Fantasy, Persona, The Elder Scrolls, and many similar games. Other franchises like Fallout add perks to this to further a sense of growth. While this is the most common method of instilling a sense of progression, it is by no means the only way to go about it. Both Deus Ex and its modern sequel, Deus Ex: Human Revolution, employed different systems. The original Deus Ex gave players Ability Points directly, after completing objectives or finding certain locations, which they could spend on skills from different types of weaponry to more passive skills such as First Aid, Lockpicking, or Swimming (which I would not recommend). Deus Ex: Human Revolution had experience, but instead of ability points which increased certain skills, they allowed protagonist Adam Jensen to unlock the cybernetic augmentations he is equipped with. Regardless of what systems are in place to encourage it, an RPG always has some way to make the player feel like his/her character is growing in either skill or power.
Another very common characteristic in RPGs is that designers tend to place a very large focus on the world and its inhabitants when making them. If players take the time to talk to people and explore in an RPG, they can expect to learn about economies, cultures, society, geography, political struggles, and more regarding the world or region that it takes place in. Games like The Elder Scrolls and Fallout (both older and newer titles) can boast a very rich and detailed world just waiting to be explored. That is one of the biggest draws of those games, and a topic I have written about before. Also, Bioware games like Mass Effect, Baldur's Gate, or Knights of the Old Republic serve as good examples. Like it or hate it, a major part of what makes the Mass Effect franchise so popular is that Bioware took the time to envision and develop a very vivid lore that most of the fans fell in love with. Learning about all of the various races, their cultures, and beliefs is half the fun of the game to some players. This is also true for the Japanese side of the RPG moniker. While games like Final Fantasy and Persona do not necessarily need to have very detailed background information due to how linear those games tend to be, players of them are often treated to pretty interesting worlds like the land of Spira in Final Fantasy X or the rural town of Inaba in Persona 4. The people and places all have there own story. The church of Yevon and the story of its creation and internal corruption are as fascinating as the discovery of a world inside the TV and all of its mysteries. When it comes down to it, all RPGs have deep, interesting worlds to learn about and/or explore.
The last element that I have noticed in all Role Playing Games to some extent is a feeling that the player has some element of choice in how the player character/party develops. Admittedly, this one is going to be a bit of a hard sell, so hear me out. In most western-style RPGs, this characteristic is pretty obvious. Usually, the player gets to choose what skills the protagonist has and/or how they develop. This is usually tied into the development system, similarly to the sense of progression. Players can often be asked at the start what class they wish to play as, a tactic employed in Alpha Protocol and other games. This can either be used separately or in combination with a system that gives players Ability Points to spend on skills as they rank up. Another well known system in Western RPGs is Skyrim's system where skills develop as they are used. From the other side of the coin, in JRPGs, this characteristic may be less noticeable, but I feel that it is still present. Games like Final Fantasy usually have characters evolve on static and fairly predictable paths, at level X they acquire ability Y. However, all of these games have some form of customization. The very first Final Fantasy allowed players to choose their character classes at the start of the game. The second had abilities level up upon using them. The third and fifth had job class systems that allowed players to experiment with different classes and truly customize their characters to their own playstyle. And most others allowed players to pick their party from a very large group. All of these games have some element that allows players to pick their own way to play through the game. The other notable JRPG, Persona, is also extremely well known for this thanks to its system where the player character and hold and use different personae while the rest of the party can be chosen from a diverse cast of character, although earlier games in the franchise allowed all party members to switch personae. Every RPG allows for players to think for themselves and play through them in their own way.
To me, all of these elements are what separate an RPG from other genres of video games. A strong sense of progression and customization along with a detailed world are ultimately what binds all of the games under this heading together. While this is the definition that I have reached, I will not claim that this definition is absolute by any means. Feel free to dispute and criticize my opinion on this subject all you like. I would welcome the conversation gladly. Whatever your own opinion is, I encourage you to discuss and share it with others.