For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique?

I’m not at all sure that there is a nice answer to the first question, but the answer to the second is no: there are non-homeomorphic spaces with homeomorphic squares. In this paper M.M. Marjanović and Ante R. Vučemilović construct a pair of non-homeomorphic countable metrizable spaces whose squares are homeomorphic.

Added: I believe that the question is originally due to S. Ulam, and that the first example is to be found in R.H. Fox, On a problem of S. Ulam concerning Cartesian products, Fund. Math. 34 (1947), 278–287; link.

Here is a necessary condition for a space to have a square root. By the Künneth theorem, if $X=Y\times Y$, then with respect to a field $F$, we must have $H_k(X;F)=\bigoplus_{i+j=k}H_i(Y;F)\otimes H_j(Y;F)$. Therefore, associating to the space $X$ the sequence of betti numbers $b_i(X;F)=\dim_F H_i(X;F)$, we must have that the sequence $(b_i)$, if consisting of finite numbers, must be the convolution square of another sequence of non-negative integers.

If there are only finitely many non-zero betti numbers for $X$, then the problem of existence of a convolution square root is the same as determining whether a polynomial has a polynomial square root with non-negative integer coefficients. If there are infinitely many non-zero betti numbers, the problem becomes one related to square roots of power series, and I'm not sure if there is an easy way to tackle that.

In any event, this gives the nice result that an odd dimensional compact manifold cannot have a square root.