There’s a very cool shortcut in Illustrator: you press Command/CTRL+d and Illustrator repeats the last action for you.
Is there a shortcut that does this or at least something similar in Photoshop?
The closest thing to this is
Edit > Transform > Again —
Command/CTRL–Shift–T — But this only works for transformations. And it only repeats the last transformation. Unlike Illustrator, it won’t do things like repeat a copy and transform.