Is there a commutative algebra over integers such that there exists $x$ with $x = y^2$ for several $y$'s?
Also, is there a commutative algebra over integers such that for every $k \in \mathbb{N}$, there exists $x$ in the algebra such that there exists a set $Y$ of cardinality $k$ of the elements $y$ that satisfy $x = y^2$?
Beside a polynomial example, would there be any monomial or numerical example?
Yes, in $R=\Bbb Z[x_j,y_{i,j}]_{i\le j}/(x_j-y_{i,j}^2)_{i\le j}$ we have $x_j=y_{i,j}^2$ for $1\le i\le j$.
You won't get any numerical example, since nothing can have $>2$ square roots in a domain.