Consider ordinary homology with coefficients in a field. For $X$ a path-connected pointed space, the graded vector space $\bigoplus_{q\ge 0} H_q(\Omega X)$ has the structure of an algebra with the multiplication induced as follows: $$ H_p(\Omega X) \otimes H_q(\Omega X) \rightarrowtail H_{p+q}(\Omega X \times \Omega X) \to H_{p+q}(\Omega X)$$ Here, the first map is a monomorphism by Kunneth's theorem. The second map is induced by the concatanetion of loops $\Omega X \times \Omega X \to \Omega X$.

Let two path-connected pointed spaces $X$ and $Y$ be given. Suppose that, for each $q\ge 0$, the two vector spaces $H_q(\Omega X)$ and $H_q(\Omega Y)$ have the same dimension. Hence $\bigoplus_{q\ge 0} H_q(\Omega X)$ and $\bigoplus_{q\ge 0} H_q(\Omega Y)$ are isomorphic as graded vector spaces. Is it necessary that $\bigoplus_{q\ge 0} H_q(\Omega X)$ and $\bigoplus_{q\ge 0} H_q(\Omega Y)$ are isomorphic as algebras?