Is it the Weapon Select Screen?
Let me explain:
From the past to the future - That would be from Mega Man to Mega Man X, which it is both present in.
I'm always here to greet you. Whenever you pause the game, want to switch a weapon, etc, this screen shows up.
Marked or anonymous. - The screen starts fairly empty when you begin a new game, until you start accumulating some weapons from bosses. In which these spaces reserved for them are either marked or anonymous.
I've been abused before. - In the early Mega Man games, bringing up the menu and then returning to the game causes Mega Man to faint-warp, which can cause some shots to go right through him undamaged.
Undamaged through the ages. - It's remained an essential means of switching weapons since the beginning, until the use of the shoulder buttons was introduced.
I usually come as twins. - Some of the Weapons you gain are divided into two menus, like in MM2 and MM3, while in some other games, like MM7, MMIV, and MMV, the first menu lists your Weapons, while the other menu usually lists Powers-Ups, Items, Bolt/P-Chip count, etc.
Chained or split. - See above.
My core remains unchanged. See first line of this passage.