You’re thinking of emblems, I believe, and I’m pretty sure that’s hardcoded to only happen in NiGHTS mode.
If you want something, enemy or otherwise, to follow the player, try using A_Chase (follows the player in a similar way to a crawla), A_DetonChase (follows the player like a deton), or any of the other actions designed to follow the player around. See the wiki page on actions in SOCs for more possible actions. I’m pretty sure you’ll have to call A_Look before doing any of these follower actions, though.