Is there another N×N matrix that works like **I**?

No. This is not quite obvious, but since 1 is unique for real numbers you might
suspect that **I** is unique for NxN matrices.

Here is why this is so.
In the following, assume that all the matrices have
the correct dimensions for the multiplications.
Suppose there is a matrix **Z** and that

ZA=A(1)

for __any__ **A** with the right dimensions.
The matrix **Z** works like **I**, but we are hoping
that it will be different.
We know how **I** works:

BI=B(2)

for __any__ **B** with the right dimensions.
Now replace **A** with **I** in (1) (since **A** can be any suitable matrix):

ZI=I(3)

Now replace **B** with **Z** in (2):

ZI=Z(4)

Looking at (3) and (4), **Z** and **I** are both equal to the
same thing, so they must equal each other:

Z=I(5)

You can start at step (1) with the other
order **AZ** = **A** and reach the same conclusion.
So if you have found a matrix that works like the indentity,
it is indeed the identity matrix.

What is the transpose of **I**?

1 0 0 0 1 0 0 0 1T