Is there another N×N matrix that works like I?
No. This is not quite obvious, but since 1 is unique you might suspect that I is unique.
It is easy to show that this is so. Suppose you had a matrix Z and that
(1) ZA = A
for any A. We know how I works:
(2) BI = B
for any B. Now use (1) with I replacing A:
(3) ZI = I
Now use (2), substituting Z for B:
(4) ZI = Z
Looking at (3) and (4):
(4) Z = I
It is also true that it works with the other order: AZ = A.