Ahhh immersion and suspension of disbelief combined. If this is what the OP user is referring to then the whole "sucking in" factor is pretty relevant when one is playing something as violent or as intense as Brutal Doom, a violent version of Doom. Sometimes, you just have to feel like you are the Doomguy and your mission is to kill all those demons by ripping their crotches out and teabagging them with their dismembered appendages. At least that's how i enjoy my experience whenever I'm playing Brutal Doom.
Also regarding the more competitive aspect of games, I remember reading about a piece about people being able to competently play a character very well because they simply like and relate well to the character. Say in Guilty Gear, i really relate to Sol Badguy, but people will tell me that if I want to be so pro in Guilty Gear (XrD) I have to use Eddie (like in all versions of Guilty Gear XX and even up to this day in XrD) or Elphett, but i tell them, screw that. In something like League of Legends, i like playing the champs Xin Zhao (hence my username here), Jax, and Jarvan IV because not only I play their skillkits very well, I also like playing as these champs since i "relate" very well to them. I can easily play carry games in solo queue far better with these champs compared to say having to play something like Elise, Gragas, Reksai, and some other top tier jungler.
I think i truly got what the OP poster is trying to ask of us.