The character select screen has appeared throughout many genres of games, although it's most recognizable for its use in fighting games.
This concept only applies to games with a list of pictures/names of all the characters you can pick from. This can also apply to games with different types of vehicles you are allowed to choose from. Most wrestling games have this concept.
The Streets of Rage series has always showed the player the stats of all of the individual characters while the Dead or Alive series often shows the characters personal stuff like height, hobby, fighting style, or even favorite food.
However, not just any game with multiple playable characters counts as a game with this concept.
In some games, like the first Marvel Vs. Capcom, there are characters that are not shown on the select screen but are still selectable by pressing an alternative button on another character, or can alternate the clothing of the displayed character ( SoulCalibur).