If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?

taking square root means reversing the effect of squaring. Dividing a number by itself does not do that (but rather always returns 1 as you noted).

Compare your question to: if doubling a number means adding it to itself, shouldn't halving a number mean subtracting it from itself? Answer: obviously not.

Squaring when explained in simple English, uses the word "itself". Here is an attempt to define the reverse process, finding square root, using the word "itself":

The square root of a number $N$ is that number $x$ such that when $N$ is divided by $x$ it gives itself (my grammar is poor, subject and object of this sentence. But I hope you get the drift)

Edit: this idea translated to an equation would give the following: if $N = 9$ then $x = 3$ and $N/x = 9$?? I guess itself in this context refers to $x$ and not $N$

Since this question hinges directly on some fundamental ideas of math, this answer attempts to explicate those ideas in a similarly fundamental way.

Squaring a number can be thought of as a procedure. The particular procedure for squaring a number can use a template like the following:

$$ \Box \longrightarrow \Box\times\Box \longrightarrow \Box $$

We put the "input" value, for example, $2$, in the leftmost box, like this:

$$ 2 \longrightarrow \Box\times\Box \longrightarrow \Box $$

Next we make copies of the leftmost box and put them in the two boxes in the middle:

$$ 2 \longrightarrow 2 \times 2 \longrightarrow \Box $$

Notice that these two boxes must each contain the same number. Finally, we perform the indicated multiplication and write the result in the last box on the right:

$$ 2 \longrightarrow 2 \times 2 \longrightarrow 4 $$

To take a square root, we want to reverse the procedure, that is, work it backwards. So we take the "input" number, for example, $9$, and put it in the box on the right:

$$ \Box \longrightarrow \Box\times\Box \longrightarrow 9 $$

Now we have to decide what to put in the two boxes in the middle. We know we need the contents of the two boxes to be equal, and we know that when we do the multiplication the result has to be $9$. Suppose we guess the number in each box should be $3$. Then we have:

$$ \Box \longrightarrow 3\times3 \longrightarrow 9 $$

We can confirm that $3\times3$ does indeed give the result $9$, so all is good so far. Now we just need to deduce what number was in the leftmost box. We know the middle boxes were filled by copying that box, so it had to contain a $3$ as well. So we have

$$ 3 \longrightarrow 3\times3 \longrightarrow 9 $$

And that's why the square root of $9$ is $3$ rather than $9/9$. (Well, that and the fact that we refuse to put $-3$ in the two boxes in the middle, because life is better when we consistently follow a rule that says a "square root" must never be a negative number.)

We may later learn how to find square roots in a way that does not rely so much on making a lucky guess. But that's a matter of an algorithm for calculating a square root, not the definition of a square root.