How can a ring of polynomials with coefficients in a field $k$, and in infinitely many variables be a subring of $k[x,y]$?

An element of $A$ can be written as a finite linear combination $\sum p_\alpha X^\alpha$ of monomials $X^\alpha$ where each monomial is of the form

$$X^\alpha = (x y^{i_1})^{j_1} \cdots (x y^{i_n})^{j_n}$$

At the end, you get an element of $k[x,y]$.

The variables of $A$, namely $x, xy, \dots$ are not independent.

I think it's a notational issue:

Let $S\le R$ be (commutative) rings, and $A=\{a_1,a_2,\dots\}\subseteq R$.
Then $S[a_1,a_2,\dots]$ simply denotes the subring generated by $S\cup A$, i.e. the smallest subring of $R$ that contains both $S$ and $A$.

Note that, however, there's a homomorphism $\varphi$ from the polynomial ring $S[x_1,x_2,\dots]$ to $R$ that maps $x_i\mapsto a_i$, and $S[a_1,a_2,\dots]$ is just the image of $\varphi$.