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 $N$