Is every Super Mario Run level iteration identical?

It is indeed hard to find a source for this, but in general, I'd back up Reinhard's answer. Most levels in World Tour mode don't contain any random components as far as I can tell after playing the game quite a lot for the last week (collecting all coloured coins in the game).

However, I can report on one level that definitely has a random component: the first Boo House level, 2-1, has small rooms with multiple green doors in each of them. One of them is the real door that leads to the next room and the other doors are fake and lead you back to the bottom of the current room. Which of the doors is the right one appears to be chosen randomly — it definitely changes between runs (and the fake doors can be recognised by their periodic shaking).

I also believe that Bowser's movement patterns are random. This is the definitely the case for the final level (which I've grinded enough for black coins to be sure). I haven't played the earlier castles often enough but it seems that his choice of jump, fireball and direction of fireball is somewhat random. For other enemies like Lakitu and Bats which don't always move the same I can't say whether they are random or whether they are just very sensitive to the player's position.