Set $X=T^2$. Denote $E$ as the square and $\partial E$ as the boundary of disk. Set $Z$ the image of $\partial E$ under quotient of $X$. I am going to abuse notation for $H_\star$ to denote both homology and reduced homology.
The book has shown $(E,\partial E)\to (X,Z)$ induces homology level isomorphism. It suffices to compute $H_\star(E,\partial E)$. I am looking for the mistake in my thought process. Consider long exact sequence $H_\star(\partial E)\to H_\star(E)\to H_\star(E,\partial E)$. Since $E$ is a square/contractible, then $H_\star(E)=0$. So $H_{\star+1}(\partial E)\cong H_\star(E,\partial E)$. This computation is clearly wrong as $H_1(X)$ should have 2 generators.
$\textbf{Q:}$ What is my mistake?
The relative homology group $H_\star(E, \partial E)$ should be giving the homology of the quotient space formed by collapsing $\partial E$ to a point. But this quotient is actually homeomorphic to the sphere $S^2$ rather than a torus. To get a torus you must instead identify opposite sides.
The homology isomorphism is giving the first homology group of $X/Z$, which is not a torus but rather a torus with two circles collapsed to a point.