Let's say I'm throwing a die and telling you whether the outcome is greater than 3 or not, then your task is to tell the conditional expectation of the outcome given this partial information.
Let's say I told you the outcome is bigger than $3$ (i.e. $X \in \{4, 5, 6\}$)... In this case the conditional expectation should be $5$
Isn't that contracting with $E(X|X)=X$?
You are confusing conditioning wrt to a random variable and conditioning wrt to an event.
Here you are conditioning wrt to an event: $$E(X | X \in \{4,5,6\})$$
Instead, this is conditioning wrt to a random variable: $$E(X | X)$$
You can look up how these objects are defined precisely, but the fact that you're confusing them is the main problem.
In particular, note that the expectation conditioned on an event is a number, while the expectation conditioned on a random variable is a random variable.
So yes, it is true that $E(X | X \in \{4,5,6\}) = 5$ and it is also true that $E(X | X) = X$. It is not true that $E(X | X \in \{4,5,6\}) = X$ (instead, it is equal to $5$ as mentioned earlier)